🧠:<think>Okay, so I have these two problems about Spanish art and history, specifically related to the Alhambra and Penrose tilings. Let me try to tackle them one by one.Starting with the first problem: The Alhambra has a tessellation pattern with regular hexagons, squares, and equilateral triangles. Each vertex is shared by one hexagon, two squares, and one triangle. I need to calculate the internal angles at each vertex and show that these polygons can tessellate the plane.Hmm, tessellation means that the shapes fit together without gaps or overlaps, right? So at each vertex where the polygons meet, the sum of their internal angles should be exactly 360 degrees. That makes sense because around a point, the full circle is 360 degrees.First, I should recall the internal angles of regular polygons. For a regular polygon with n sides, the internal angle is given by the formula:[text{Internal angle} = frac{(n - 2) times 180^circ}{n}]So, let's calculate the internal angles for each shape:1. Regular Hexagon (6 sides):[frac{(6 - 2) times 180^circ}{6} = frac{4 times 180^circ}{6} = frac{720^circ}{6} = 120^circ]2. Square (4 sides):[frac{(4 - 2) times 180^circ}{4} = frac{2 times 180^circ}{4} = frac{360^circ}{4} = 90^circ]3. Equilateral Triangle (3 sides):[frac{(3 - 2) times 180^circ}{3} = frac{1 times 180^circ}{3} = frac{180^circ}{3} = 60^circ]Okay, so each hexagon contributes 120 degrees, each square contributes 90 degrees, and each triangle contributes 60 degrees at the vertex.Now, according to the problem, each vertex is shared by one hexagon, two squares, and one triangle. So, let's add up their angles:- One hexagon: 120°- Two squares: 2 × 90° = 180°- One triangle: 60°Total angle at the vertex: 120° + 180° + 60° = 360°Perfect! That adds up to exactly 360 degrees, which means the polygons can fit together perfectly around each vertex without any gaps or overlaps. Therefore, this combination of regular hexagons, squares, and equilateral triangles can tessellate the plane.Wait, just to make sure I didn't make a mistake. Let me double-check the internal angles:- Hexagon: 120°, correct.- Square: 90°, correct.- Triangle: 60°, correct.Adding them: 120 + 90 + 90 + 60. Wait, hold on, is it one hexagon, two squares, and one triangle? So that's 120 + 90 + 90 + 60. Wait, that's four polygons? Or is it one hexagon, two squares, and one triangle, so that's four polygons in total at each vertex? Let me see.Wait, no, the problem says each vertex is shared by one hexagon, two squares, and one triangle. So that's 1 + 2 + 1 = 4 polygons meeting at each vertex. So, the angles are 120 + 90 + 90 + 60, which is indeed 360. So that's correct.So, yeah, the internal angles at each vertex sum up to 360°, so the tessellation is possible.Moving on to the second problem: Inspired by the geometric patterns, I want to create a complex mosaic design based on a Penrose tiling. Penrose tilings are non-periodic, meaning they don't repeat in a regular pattern, and they're generated by two shapes: kites and darts. The ratio of kites to darts is given by the golden ratio, φ, where φ = (1 + √5)/2. If I have 100 kites, how many darts are present?Alright, so the ratio of kites to darts is φ. So, kites:darts = φ:1. That means for every φ kites, there is 1 dart. Alternatively, the number of darts is the number of kites divided by φ.Given that I have 100 kites, so the number of darts should be 100 / φ.But let me make sure. The ratio is kites to darts is φ, so kites/darts = φ. So, darts = kites / φ.Yes, that seems right.So, φ is (1 + √5)/2, approximately 1.618.So, number of darts = 100 / φ ≈ 100 / 1.618 ≈ 61.8.But since we can't have a fraction of a dart, we need to see if the number is an integer. But in Penrose tilings, the ratio is exact, so perhaps it's an exact number.Wait, actually, in Penrose tilings, the ratio of kites to darts is indeed the golden ratio, but it's an irrational number, so in a finite tiling, you can't have an exact ratio. However, in the limit as the tiling becomes large, the ratio approaches φ.But in this case, the problem says the ratio is given by φ, so perhaps we can take it as exact.So, if kites:darts = φ:1, then darts = kites / φ.Given that, with 100 kites, darts = 100 / φ.But φ is (1 + √5)/2, so 1/φ is 2/(1 + √5). Let's rationalize the denominator:1/φ = 2/(1 + √5) = [2(1 - √5)] / [(1 + √5)(1 - √5)] = [2(1 - √5)] / (1 - 5) = [2(1 - √5)] / (-4) = [2(√5 - 1)] / 4 = (√5 - 1)/2.So, 1/φ = (√5 - 1)/2 ≈ (2.236 - 1)/2 ≈ 1.236/2 ≈ 0.618.Therefore, number of darts = 100 × (√5 - 1)/2 ≈ 100 × 0.618 ≈ 61.8.But since we can't have a fraction, maybe we need to round it? Or perhaps the problem expects an exact value in terms of φ.Wait, let me think again. The ratio is kites:darts = φ:1, so kites = φ × darts.Therefore, darts = kites / φ.Given kites = 100, so darts = 100 / φ.But 100 / φ is equal to 100 × (2)/(1 + √5) = 200 / (1 + √5). Rationalizing the denominator:200 / (1 + √5) × (1 - √5)/(1 - √5) = 200(1 - √5)/(1 - 5) = 200(1 - √5)/(-4) = -50(1 - √5) = 50(√5 - 1).So, darts = 50(√5 - 1).Calculating that numerically: √5 ≈ 2.236, so √5 - 1 ≈ 1.236. Therefore, 50 × 1.236 ≈ 61.8.So, approximately 61.8 darts. But since we can't have a fraction, perhaps the problem expects an exact expression or maybe it's okay to have a fractional number in the context of tiling, but in reality, you can't have a fraction of a tile.Wait, but in Penrose tilings, the ratio is maintained asymptotically, so in a finite tiling, you can have an approximate number. But the problem says "the ratio of kites to darts is given by the golden ratio," so maybe it's expecting an exact value in terms of φ.But if we express darts as 100 / φ, which is 100 × (2)/(1 + √5), which simplifies to 50(√5 - 1), as above.Alternatively, since φ = (1 + √5)/2, then 1/φ = (√5 - 1)/2, so darts = 100 × (√5 - 1)/2 = 50(√5 - 1).So, maybe the answer is 50(√5 - 1), which is approximately 61.8, but since we can't have a fraction, perhaps it's 62 darts? Or maybe the problem expects the exact expression.Wait, let me check the problem statement again: "the ratio of kites to darts in a Penrose tiling is given by the golden ratio, φ... and you have 100 kites, how many darts are present in your design?"So, it's asking for the number of darts, given 100 kites and the ratio kites:darts = φ:1.So, mathematically, darts = kites / φ = 100 / φ.But 100 / φ is equal to 100 × (2)/(1 + √5) = 200 / (1 + √5). Rationalizing, as above, gives 50(√5 - 1).So, 50(√5 - 1) is the exact number. If we compute that, it's approximately 61.8, but since we can't have a fraction, perhaps we need to express it as 50(√5 - 1) or approximate it.But in the context of the problem, since it's a design, maybe it's acceptable to have a fractional number, or perhaps it's expecting the exact expression.Alternatively, maybe the ratio is kites to darts is 1:φ, but no, the problem says "the ratio of kites to darts is given by the golden ratio, φ," so kites:darts = φ:1.So, kites/darts = φ, so darts = kites / φ.Therefore, the exact number is 100 / φ, which is 50(√5 - 1). So, I think that's the answer they're expecting.Alternatively, if we rationalize, 50(√5 - 1) is the exact value, which is approximately 61.8, but since we can't have a fraction, maybe we need to round it. But in the context of Penrose tilings, which are aperiodic and can have irrational ratios, perhaps the exact value is acceptable.So, I think the answer is 50(√5 - 1), which is approximately 61.8, but since it's a mathematical problem, they might prefer the exact form.Alternatively, maybe the ratio is kites:darts = φ:1, so if kites = 100, then darts = 100 / φ = 100 × (2)/(1 + √5) = 200 / (1 + √5). Rationalizing, we get 50(√5 - 1). So, that's the exact number.Therefore, the number of darts is 50(√5 - 1), which is approximately 61.8, but since we can't have a fraction, perhaps the problem expects the exact expression.Wait, but in the problem statement, it's about a design, so maybe it's okay to have a fractional number? Or perhaps it's a theoretical question, so the exact value is fine.So, to sum up, the number of darts is 100 divided by φ, which is 50(√5 - 1). So, I think that's the answer.But let me double-check the ratio. If kites:darts = φ:1, then kites = φ × darts. So, darts = kites / φ. So, with 100 kites, darts = 100 / φ.Yes, that's correct.So, the exact number is 50(√5 - 1), which is approximately 61.8, but since we can't have a fraction, maybe the problem expects the exact form.Alternatively, perhaps the ratio is kites:darts = 1:φ, but that would mean darts = kites × φ, which would be more darts than kites, but the problem says the ratio is given by φ, so kites:darts = φ:1.So, I think the answer is 50(√5 - 1), which is approximately 61.8, but since it's a mathematical problem, the exact form is better.So, I think that's it.Final Answer1. The internal angles at each vertex sum to boxed{360^circ}, confirming the tessellation.2. The number of darts is boxed{50(sqrt{5} - 1)}.</think>

🧠:<think>Okay, so I have this problem where a business tycoon wants to expand into three regions: A, B, and C. He needs to figure out how many facilities to open in each region to minimize the total regulatory cost. Then, he also has to make sure that the total risk factor doesn't exceed 1.5. Hmm, sounds like an optimization problem with constraints. Let me try to break it down step by step.First, let's tackle the first part. The regulatory costs for each region are given by quadratic functions:- Region A: ( C_A(x) = 3x^2 + 5x + 2 )- Region B: ( C_B(x) = 4x^2 + 6x + 3 )- Region C: ( C_C(x) = 5x^2 + 4x + 1 )And the total number of facilities is 15, so ( x_A + x_B + x_C = 15 ). The goal is to minimize the total cost, which is the sum of the costs from each region: ( C_A + C_B + C_C ).Since this is a constrained optimization problem, I think I should use the method of Lagrange multipliers. That method helps find the local maxima and minima of a function subject to equality constraints. So, I need to set up the Lagrangian function.Let me denote the total cost as ( C = C_A + C_B + C_C ). So,( C = 3x_A^2 + 5x_A + 2 + 4x_B^2 + 6x_B + 3 + 5x_C^2 + 4x_C + 1 )Simplify that:( C = 3x_A^2 + 4x_B^2 + 5x_C^2 + 5x_A + 6x_B + 4x_C + 6 )The constraint is ( x_A + x_B + x_C = 15 ). So, the Lagrangian function ( mathcal{L} ) is:( mathcal{L} = 3x_A^2 + 4x_B^2 + 5x_C^2 + 5x_A + 6x_B + 4x_C + 6 + lambda(15 - x_A - x_B - x_C) )Wait, actually, the Lagrangian is the function to be optimized minus lambda times the constraint. But since we're minimizing, it's the same as adding lambda times the constraint. Hmm, actually, I think it's:( mathcal{L} = 3x_A^2 + 4x_B^2 + 5x_C^2 + 5x_A + 6x_B + 4x_C + 6 + lambda(15 - x_A - x_B - x_C) )Yes, that seems right. Now, to find the minimum, we take the partial derivatives of ( mathcal{L} ) with respect to each variable ( x_A, x_B, x_C, lambda ) and set them equal to zero.So, let's compute the partial derivatives.First, partial derivative with respect to ( x_A ):( frac{partial mathcal{L}}{partial x_A} = 6x_A + 5 - lambda = 0 )Similarly, partial derivative with respect to ( x_B ):( frac{partial mathcal{L}}{partial x_B} = 8x_B + 6 - lambda = 0 )Partial derivative with respect to ( x_C ):( frac{partial mathcal{L}}{partial x_C} = 10x_C + 4 - lambda = 0 )And partial derivative with respect to ( lambda ):( frac{partial mathcal{L}}{partial lambda} = 15 - x_A - x_B - x_C = 0 )So now, we have a system of four equations:1. ( 6x_A + 5 = lambda )2. ( 8x_B + 6 = lambda )3. ( 10x_C + 4 = lambda )4. ( x_A + x_B + x_C = 15 )Our goal is to solve for ( x_A, x_B, x_C ). Let me express each ( x ) in terms of ( lambda ).From equation 1:( 6x_A = lambda - 5 ) => ( x_A = (lambda - 5)/6 )From equation 2:( 8x_B = lambda - 6 ) => ( x_B = (lambda - 6)/8 )From equation 3:( 10x_C = lambda - 4 ) => ( x_C = (lambda - 4)/10 )Now, substitute these expressions into equation 4:( (lambda - 5)/6 + (lambda - 6)/8 + (lambda - 4)/10 = 15 )Let me compute each term:First term: ( (lambda - 5)/6 )Second term: ( (lambda - 6)/8 )Third term: ( (lambda - 4)/10 )To add these fractions, I need a common denominator. Let's find the least common multiple (LCM) of 6, 8, and 10. The prime factors are:- 6: 2, 3- 8: 2^3- 10: 2, 5So, LCM is 2^3 * 3 * 5 = 8 * 3 * 5 = 120.Convert each fraction to have denominator 120:First term: ( (lambda - 5)/6 = 20(lambda - 5)/120 )Second term: ( (lambda - 6)/8 = 15(lambda - 6)/120 )Third term: ( (lambda - 4)/10 = 12(lambda - 4)/120 )So, adding them up:[20(λ - 5) + 15(λ - 6) + 12(λ - 4)] / 120 = 15Multiply both sides by 120:20(λ - 5) + 15(λ - 6) + 12(λ - 4) = 15 * 120Compute each term:20λ - 100 + 15λ - 90 + 12λ - 48 = 1800Combine like terms:(20λ + 15λ + 12λ) + (-100 - 90 - 48) = 180047λ - 238 = 1800Add 238 to both sides:47λ = 1800 + 238 = 2038So, λ = 2038 / 47Let me compute that:47 * 43 = 2021 (since 47*40=1880, 47*3=141; 1880+141=2021)2038 - 2021 = 17So, λ = 43 + 17/47 ≈ 43.3617So, λ ≈ 43.3617Now, let's find each x:x_A = (λ - 5)/6 ≈ (43.3617 - 5)/6 ≈ 38.3617 / 6 ≈ 6.3936x_B = (λ - 6)/8 ≈ (43.3617 - 6)/8 ≈ 37.3617 / 8 ≈ 4.6702x_C = (λ - 4)/10 ≈ (43.3617 - 4)/10 ≈ 39.3617 / 10 ≈ 3.9362Hmm, so x_A ≈ 6.3936, x_B ≈ 4.6702, x_C ≈ 3.9362But the number of facilities must be integers, right? Because you can't have a fraction of a facility. So, this complicates things a bit. The Lagrange multipliers method gives us a continuous solution, but in reality, we need integer values.But maybe the problem allows for fractional facilities? It doesn't specify, so perhaps we can assume that x_A, x_B, x_C can be real numbers, and we can just take the approximate decimal values. Alternatively, maybe we need to round them to the nearest integer and check if the total is 15.Let me see. If I round x_A to 6, x_B to 5, and x_C to 4, then total is 6+5+4=15, which is good.But wait, let's check the exact fractions.Wait, λ = 2038 / 47. Let me compute that exactly.2038 divided by 47:47*43 = 2021, as above.2038 - 2021 = 17, so 2038 / 47 = 43 + 17/47.So, 17/47 is approximately 0.3617.So, x_A = (43 + 17/47 - 5)/6 = (38 + 17/47)/6 = (38*47 +17)/47 /6Wait, maybe better to compute numerically:x_A = (43.3617 - 5)/6 = 38.3617 /6 ≈6.3936x_B = (43.3617 -6)/8≈37.3617 /8≈4.6702x_C=(43.3617 -4)/10≈39.3617 /10≈3.9362So, approximately, x_A≈6.39, x_B≈4.67, x_C≈3.94.So, if we need integer values, we can try different combinations around these numbers.But perhaps the problem expects us to just use the real numbers, even though in reality, you can't have a fraction of a facility. Maybe it's just a mathematical solution.Alternatively, perhaps we can use the exact fractions.Wait, let's see:x_A = (λ -5)/6 = (2038/47 -5)/6 = (2038 - 235)/47 /6 = (1803)/47 /6 = 1803/(47*6)=1803/282=6.3936...Similarly, x_B=(2038/47 -6)/8=(2038 - 282)/47 /8=1756/47 /8=1756/(47*8)=1756/376≈4.6702x_C=(2038/47 -4)/10=(2038 - 188)/47 /10=1850/47 /10=1850/(47*10)=1850/470≈3.9362So, exact fractions are 1803/282, 1756/376, 1850/470.Simplify:1803/282: divide numerator and denominator by 3: 601/94 ≈6.39361756/376: divide numerator and denominator by 4: 439/94≈4.67021850/470: divide numerator and denominator by 10: 185/47≈3.9362So, these are the exact values.But since the problem doesn't specify whether x must be integers, perhaps we can just present these fractional values as the solution.But let me check the problem statement again: "the optimal number of facilities x_A, x_B, and x_C to be opened in each region." It doesn't specify integer, so maybe fractional is acceptable.Alternatively, perhaps the problem expects us to use the continuous solution and then round, but I think in optimization, unless specified, fractional solutions are acceptable, especially since it's a mathematical model.So, moving forward, the optimal solution is approximately x_A≈6.39, x_B≈4.67, x_C≈3.94.But let me verify if these satisfy the constraint x_A +x_B +x_C=15.6.39 +4.67 +3.94≈15.00, so yes, approximately 15.So, that's the first part.Now, moving on to the second part. The risk factors for each region are given by:- Region A: ( R_A(x) = frac{2}{x+1} )- Region B: ( R_B(x) = frac{3}{x+2} )- Region C: ( R_C(x) = frac{4}{x+3} )The total risk factor should not exceed 1.5. So, the constraint is:( R_A + R_B + R_C leq 1.5 )Using the solution from the first part, let's compute the total risk.So, plug in x_A≈6.39, x_B≈4.67, x_C≈3.94.Compute R_A: 2/(6.39 +1)=2/7.39≈0.2706R_B:3/(4.67 +2)=3/6.67≈0.4496R_C:4/(3.94 +3)=4/6.94≈0.5764Total risk≈0.2706 +0.4496 +0.5764≈1.2966Which is approximately 1.2966, which is less than 1.5. So, the solution satisfies the risk constraint.But wait, let me compute it more accurately.Compute R_A: 2/(6.3936 +1)=2/7.3936≈0.2706R_B:3/(4.6702 +2)=3/6.6702≈0.4496R_C:4/(3.9362 +3)=4/6.9362≈0.5764Adding them up: 0.2706 +0.4496 =0.7202; 0.7202 +0.5764≈1.2966So, total risk≈1.2966, which is below 1.5. Therefore, the solution from the first part satisfies the risk constraint.But just to be thorough, let me check if the exact fractions give the same result.Compute R_A: 2/(x_A +1)=2/(6.3936 +1)=2/7.3936≈0.2706Similarly, R_B and R_C as above.So, same result.Therefore, the solution from the first part is acceptable in terms of risk.But wait, just to make sure, let me compute the exact fractions.x_A=6.3936=6 + 0.3936≈6 + 25/63≈6.3968, but maybe better to keep it as 6.3936.Alternatively, using exact fractions:x_A=601/94≈6.3936x_B=439/94≈4.6702x_C=185/47≈3.9362Compute R_A=2/(x_A +1)=2/(601/94 +1)=2/(601/94 +94/94)=2/(695/94)=2*(94/695)=188/695≈0.2706R_B=3/(x_B +2)=3/(439/94 +2)=3/(439/94 +188/94)=3/(627/94)=3*(94/627)=282/627≈0.4496R_C=4/(x_C +3)=4/(185/47 +3)=4/(185/47 +141/47)=4/(326/47)=4*(47/326)=188/326≈0.5764So, total risk=188/695 +282/627 +188/326Let me compute each fraction:188/695≈0.2706282/627≈0.4496188/326≈0.5764Adding up:≈0.2706 +0.4496 +0.5764≈1.2966So, same as before.Therefore, the total risk is approximately 1.2966, which is less than 1.5, so the solution satisfies the risk constraint.Therefore, no adjustments are needed.But just to be thorough, let me check if the risk is indeed less than 1.5. Since 1.2966 <1.5, yes, it's satisfied.So, the optimal number of facilities is approximately x_A≈6.39, x_B≈4.67, x_C≈3.94, which sum to 15, and the total risk is approximately 1.2966, which is below the threshold of 1.5.Therefore, the solution is acceptable.But wait, just to make sure, let me think if there's any other consideration. For example, if the tycoon wants to minimize cost, but also wants to ensure risk is below 1.5. Since the initial solution already satisfies the risk constraint, there's no need to adjust. If it didn't, we would have to incorporate the risk constraint into the optimization problem, perhaps using another Lagrangian with two constraints, but in this case, it's not necessary.So, in conclusion, the optimal number of facilities is approximately 6.39 in A, 4.67 in B, and 3.94 in C, and the total risk is about 1.2966, which is within the acceptable limit.Final AnswerThe optimal number of facilities to be opened in each region are approximately ( boxed{6.39} ) in Region A, ( boxed{4.67} ) in Region B, and ( boxed{3.94} ) in Region C. The total risk factor is approximately 1.2966, which satisfies the constraint of not exceeding 1.5.</think>

🧠:<think>Okay, so I have this problem about the Sophomore's Dream function, which is defined as the integral ( S(x) = int_0^x t^{-t} , dt ). There are two parts: first, I need to compute ( S(1) ) and express it in terms of a known mathematical constant or a series. Second, I have to explore the connection between this function and number theory by looking at ( f(n) = sum_{k=1}^n k^{-k} ) and determine if it converges as ( n ) approaches infinity, and if so, find its limit.Starting with the first part, computing ( S(1) ). So, ( S(1) = int_0^1 t^{-t} , dt ). Hmm, I remember that ( t^{-t} ) can be expressed using exponentials. Let me recall that ( t^{-t} = e^{-t ln t} ). That might be helpful because integrating exponentials can sometimes be done using series expansions.So, let's write ( t^{-t} = e^{-t ln t} ). Then, the integral becomes ( int_0^1 e^{-t ln t} , dt ). Now, I remember that for small values of ( x ), ( e^x ) can be expanded as a Taylor series: ( e^x = sum_{n=0}^infty frac{x^n}{n!} ). Maybe I can use this expansion for ( e^{-t ln t} ).Let me try that. So, substituting ( x = -t ln t ), we have:( e^{-t ln t} = sum_{n=0}^infty frac{(-t ln t)^n}{n!} ).Therefore, the integral ( S(1) ) becomes:( S(1) = int_0^1 sum_{n=0}^infty frac{(-t ln t)^n}{n!} , dt ).Assuming I can interchange the integral and the summation (which I think is valid here because of uniform convergence, but I should double-check that later), this becomes:( S(1) = sum_{n=0}^infty frac{(-1)^n}{n!} int_0^1 t^n (ln t)^n , dt ).Hmm, okay, so now I have an integral of the form ( int_0^1 t^n (ln t)^n , dt ). I need to compute this integral. Let me make a substitution to simplify it. Let me set ( u = -ln t ). Then, when ( t = 1 ), ( u = 0 ), and when ( t = 0 ), ( u ) approaches infinity. Also, ( dt = -e^{-u} du ).So, substituting, the integral becomes:( int_{infty}^0 (e^{-u})^n (-u)^n (-e^{-u} du) ).Wait, let's be careful with the substitution. So, ( t = e^{-u} ), so ( ln t = -u ), and ( dt = -e^{-u} du ). So, substituting into the integral:( int_0^1 t^n (ln t)^n , dt = int_{infty}^0 (e^{-u})^n (-u)^n (-e^{-u} du) ).Simplify the expression step by step. First, ( t^n = e^{-n u} ), ( (ln t)^n = (-u)^n ), and ( dt = -e^{-u} du ). So, putting it all together:( int_{infty}^0 e^{-n u} (-u)^n (-e^{-u}) du ).Let me handle the signs and the limits. The integral from ( infty ) to 0 can be rewritten as the negative integral from 0 to ( infty ). So:( - int_0^infty e^{-n u} (-u)^n (-e^{-u}) du ).Wait, let me count the negative signs. The substitution gives ( dt = -e^{-u} du ), so the integral becomes:( int_{infty}^0 e^{-n u} (-u)^n (-e^{-u}) du ).Which is:( int_{infty}^0 e^{-n u} (-1)^n u^n (-e^{-u}) du ).Simplify the terms:The two negative signs from ( (-u)^n ) and ( dt ) give ( (-1)^n times (-1) = (-1)^{n+1} ). So, the integral becomes:( (-1)^{n+1} int_{infty}^0 e^{-(n+1)u} u^n du ).But integrating from ( infty ) to 0 is the same as integrating from 0 to ( infty ) with a negative sign:( (-1)^{n+1} times (-1) int_0^infty e^{-(n+1)u} u^n du ).Which simplifies to:( (-1)^n int_0^infty e^{-(n+1)u} u^n du ).Now, the integral ( int_0^infty e^{-(n+1)u} u^n du ) is a standard Gamma function integral. Recall that ( Gamma(k) = int_0^infty e^{-x} x^{k-1} dx ). So, if we make a substitution ( x = (n+1)u ), then ( u = x/(n+1) ), and ( du = dx/(n+1) ). Substituting, the integral becomes:( int_0^infty e^{-x} left( frac{x}{n+1} right)^n frac{dx}{n+1} ).Which is:( frac{1}{(n+1)^{n+1}} int_0^infty e^{-x} x^n dx ).And ( int_0^infty e^{-x} x^n dx = Gamma(n+1) ). Since ( n ) is an integer, ( Gamma(n+1) = n! ). Therefore, the integral becomes:( frac{n!}{(n+1)^{n+1}} ).So, putting it all together, the integral ( int_0^1 t^n (ln t)^n dt = (-1)^n times frac{n!}{(n+1)^{n+1}} ).Wait, hold on. Let me recap:We had:( int_0^1 t^n (ln t)^n dt = (-1)^n times frac{n!}{(n+1)^{n+1}} ).But wait, let me check the substitution again because I might have messed up the signs somewhere.Wait, when I did the substitution ( u = -ln t ), then ( t = e^{-u} ), ( dt = -e^{-u} du ). So, when ( t ) goes from 0 to 1, ( u ) goes from ( infty ) to 0. So, the integral becomes:( int_{infty}^0 (e^{-u})^n (-u)^n (-e^{-u}) du ).Let me write that as:( int_{infty}^0 e^{-n u} (-u)^n (-e^{-u}) du ).Which is:( int_{infty}^0 e^{-(n+1)u} (-u)^n du ).Now, ( (-u)^n = (-1)^n u^n ), so:( (-1)^n int_{infty}^0 e^{-(n+1)u} u^n du ).Changing the limits from ( infty ) to 0 to 0 to ( infty ) introduces a negative sign:( (-1)^n times (-1) int_0^infty e^{-(n+1)u} u^n du ).Which is:( (-1)^{n+1} int_0^infty e^{-(n+1)u} u^n du ).Then, as before, substituting ( x = (n+1)u ), ( u = x/(n+1) ), ( du = dx/(n+1) ):( (-1)^{n+1} times frac{1}{(n+1)^{n+1}} int_0^infty e^{-x} x^n dx ).Which is:( (-1)^{n+1} times frac{n!}{(n+1)^{n+1}} ).So, putting it all together:( int_0^1 t^n (ln t)^n dt = (-1)^{n+1} times frac{n!}{(n+1)^{n+1}} ).Wait, so that's different from what I had before. So, in the expression for ( S(1) ), we have:( S(1) = sum_{n=0}^infty frac{(-1)^n}{n!} times left( (-1)^{n+1} frac{n!}{(n+1)^{n+1}} right) ).Simplify this:The ( (-1)^n ) from the series and the ( (-1)^{n+1} ) from the integral multiply to ( (-1)^{2n+1} = (-1)^{1} = -1 ).Also, ( n! ) cancels with ( 1/n! ), so we're left with:( S(1) = sum_{n=0}^infty frac{ -1 }{ (n+1)^{n+1} } ).But wait, let's write that out:( S(1) = - sum_{n=0}^infty frac{1}{(n+1)^{n+1}} ).But notice that when ( n = 0 ), the term is ( -1/(1^{1}) = -1 ). Then, for ( n = 1 ), it's ( -1/(2^2) = -1/4 ), and so on.But wait, let's think about the original integral ( S(1) = int_0^1 t^{-t} dt ). Since ( t^{-t} ) is always positive on (0,1), the integral should be positive. However, according to this, ( S(1) ) is negative, which doesn't make sense. So, I must have messed up the sign somewhere.Let me go back through the substitution steps.Starting again, ( u = -ln t ), so ( t = e^{-u} ), ( dt = -e^{-u} du ). So, when ( t = 0 ), ( u = infty ); when ( t = 1 ), ( u = 0 ). So, the integral becomes:( int_{infty}^0 e^{-n u} (-u)^n (-e^{-u}) du ).Breaking it down:- ( t^n = (e^{-u})^n = e^{-n u} )- ( (ln t)^n = (-u)^n )- ( dt = -e^{-u} du )So, putting it all together:( int_{infty}^0 e^{-n u} (-u)^n (-e^{-u}) du ).Simplify the terms:- The first term is ( e^{-n u} )- The second term is ( (-u)^n = (-1)^n u^n )- The third term is ( -e^{-u} du )So, combining all:( int_{infty}^0 e^{-n u} (-1)^n u^n (-e^{-u}) du ).Multiplying the constants:( (-1)^n times (-1) = (-1)^{n+1} )So, the integral becomes:( (-1)^{n+1} int_{infty}^0 e^{-(n+1)u} u^n du ).Changing the limits from ( infty ) to 0 to 0 to ( infty ) introduces another negative sign:( (-1)^{n+1} times (-1) int_0^infty e^{-(n+1)u} u^n du ).Which is:( (-1)^n int_0^infty e^{-(n+1)u} u^n du ).Then, substituting ( x = (n+1)u ), we get:( (-1)^n times frac{1}{(n+1)^{n+1}} int_0^infty e^{-x} x^n dx ).Which is:( (-1)^n times frac{n!}{(n+1)^{n+1}} ).So, the integral ( int_0^1 t^n (ln t)^n dt = (-1)^n times frac{n!}{(n+1)^{n+1}} ).Wait, so previously, I had an extra negative sign, but now I think this is correct.So, going back to ( S(1) ):( S(1) = sum_{n=0}^infty frac{(-1)^n}{n!} times left( (-1)^n frac{n!}{(n+1)^{n+1}} right) ).Simplify the terms:The ( (-1)^n ) from the series and the ( (-1)^n ) from the integral multiply to ( (-1)^{2n} = 1 ).The ( n! ) cancels with ( 1/n! ), so we're left with:( S(1) = sum_{n=0}^infty frac{1}{(n+1)^{n+1}} ).Ah, that makes more sense because all terms are positive, so the integral is positive, which aligns with our expectation.So, ( S(1) = sum_{n=0}^infty frac{1}{(n+1)^{n+1}} ).But let's write it in a more standard form. Let me shift the index by letting ( k = n + 1 ). Then, when ( n = 0 ), ( k = 1 ), and as ( n ) approaches infinity, ( k ) approaches infinity. So, the sum becomes:( S(1) = sum_{k=1}^infty frac{1}{k^{k}} ).Ah, so that's the series representation of ( S(1) ). I think this is known as the Sophomore's Dream, which is a famous identity in calculus. So, ( S(1) ) is equal to the sum from ( k = 1 ) to infinity of ( 1/k^k ).I remember that this sum converges to approximately 1.291285997... which is a known constant, but it doesn't have a simpler closed-form expression in terms of elementary functions. So, the answer to part 1 is that ( S(1) ) is equal to this infinite series ( sum_{k=1}^infty frac{1}{k^k} ).Moving on to part 2, we need to explore the connection between the Sophomore's Dream function and number theory by investigating ( f(n) = sum_{k=1}^n k^{-k} ). We need to prove if ( f(n) ) converges as ( n ) approaches infinity, and if so, determine its limit.Well, from part 1, we already saw that ( S(1) = sum_{k=1}^infty frac{1}{k^k} ). So, ( f(n) ) is just the partial sum of this series up to ( n ). Therefore, as ( n ) approaches infinity, ( f(n) ) approaches ( S(1) ), which is the Sophomore's Dream constant.But to make this rigorous, we need to show that the series ( sum_{k=1}^infty frac{1}{k^k} ) converges. Since each term ( 1/k^k ) is positive, we can use the comparison test or root test or ratio test.Let me consider the root test. The root test says that for a series ( sum a_k ), compute ( limsup_{k to infty} sqrt[k]{|a_k|} ). If this limit is less than 1, the series converges absolutely.Here, ( a_k = 1/k^k ), so ( sqrt[k]{a_k} = sqrt[k]{1/k^k} = 1/k ). As ( k ) approaches infinity, ( 1/k ) approaches 0, which is less than 1. Therefore, by the root test, the series converges absolutely.Alternatively, we can use the comparison test. For ( k geq 1 ), ( k^k geq 2^{k} ) for ( k geq 2 ). Wait, is that true? Let's check for ( k = 2 ): ( 2^2 = 4 ) and ( 2^2 = 4 ), equal. For ( k = 3 ): ( 3^3 = 27 ) vs ( 2^3 = 8 ), so yes, ( k^k geq 2^k ) for ( k geq 2 ).Therefore, ( 1/k^k leq 1/2^k ) for ( k geq 2 ). The series ( sum_{k=2}^infty 1/2^k ) is a geometric series with ratio 1/2, which converges. Therefore, by the comparison test, ( sum_{k=2}^infty 1/k^k ) converges, and since the first term ( 1/1^1 = 1 ) is finite, the entire series ( sum_{k=1}^infty 1/k^k ) converges.Therefore, ( f(n) ) converges as ( n ) approaches infinity, and its limit is ( S(1) ), which is the Sophomore's Dream constant ( sum_{k=1}^infty 1/k^k ).So, summarizing:1. ( S(1) = sum_{k=1}^infty frac{1}{k^k} ).2. ( f(n) = sum_{k=1}^n k^{-k} ) converges to ( S(1) ) as ( n ) approaches infinity.I think that covers both parts of the problem. I should double-check if there's any step where I might have made a mistake, especially with the signs in the integral substitution, but it seems consistent now because the final result for ( S(1) ) is positive, which matches the integral of a positive function.Also, regarding the convergence of ( f(n) ), since the series converges, the partial sums ( f(n) ) must approach the sum of the series, which is ( S(1) ).Final Answer1. ( S(1) ) is equal to the series ( sum_{k=1}^infty frac{1}{k^k} ), so the result is boxed{sum_{k=1}^infty frac{1}{k^k}}.2. The function ( f(n) ) converges as ( n ) approaches infinity, and its limit is the same series, so the limit is boxed{sum_{k=1}^infty frac{1}{k^k}}.</think>

🧠:<think>Okay, so I have this problem about a stochastic differential equation (SDE) modeling the transcription rate of mRNA in bacteria. The equation given is:[ dX(t) = alpha X(t) dt + sigma X(t) dW(t) ]where ( X(t) ) is the mRNA concentration, ( alpha ) and ( sigma ) are constants, and ( W(t) ) is a standard Wiener process. The initial condition is ( X(0) = X_0 ).I need to find the expected value ( E[X(t)] ) and the variance ( text{Var}(X(t)) ) of the mRNA concentration at time ( t ).Alright, let me start by recalling what I know about SDEs. This equation looks like a geometric Brownian motion, which is a common model in finance and biology. The general form is:[ dX(t) = mu X(t) dt + sigma X(t) dW(t) ]Comparing this to the given equation, it seems like ( mu ) here is ( alpha ). So, this is indeed a geometric Brownian motion.For geometric Brownian motion, I remember that the solution can be written in terms of exponentials. Let me try to write down the solution. The solution to the SDE is:[ X(t) = X(0) expleft( left( alpha - frac{sigma^2}{2} right) t + sigma W(t) right) ]Wait, is that right? Let me think. The standard solution for a geometric Brownian motion is:[ X(t) = X(0) expleft( (mu - frac{sigma^2}{2}) t + sigma W(t) right) ]Yes, so in our case, ( mu = alpha ), so substituting, we get:[ X(t) = X_0 expleft( left( alpha - frac{sigma^2}{2} right) t + sigma W(t) right) ]Okay, so that's the solution. Now, to find the expected value ( E[X(t)] ), I need to compute the expectation of this expression.So, ( E[X(t)] = Eleft[ X_0 expleft( left( alpha - frac{sigma^2}{2} right) t + sigma W(t) right) right] ).Since ( X_0 ) is a constant (given as the initial condition), I can factor that out:( E[X(t)] = X_0 Eleft[ expleft( left( alpha - frac{sigma^2}{2} right) t + sigma W(t) right) right] ).Now, the expectation of the exponential of a normal random variable. I remember that if ( W(t) ) is a standard Brownian motion, then ( W(t) ) is normally distributed with mean 0 and variance ( t ). So, ( W(t) sim N(0, t) ).Let me denote ( Y = left( alpha - frac{sigma^2}{2} right) t + sigma W(t) ). Then, ( Y ) is a linear transformation of a normal variable, so it's also normal. Let's compute its mean and variance.The mean of ( Y ) is ( E[Y] = left( alpha - frac{sigma^2}{2} right) t + sigma E[W(t)] ). Since ( E[W(t)] = 0 ), this simplifies to ( E[Y] = left( alpha - frac{sigma^2}{2} right) t ).The variance of ( Y ) is ( text{Var}(Y) = text{Var}left( left( alpha - frac{sigma^2}{2} right) t + sigma W(t) right) ). Since ( left( alpha - frac{sigma^2}{2} right) t ) is a constant, its variance is zero. Therefore, ( text{Var}(Y) = sigma^2 text{Var}(W(t)) ). Since ( text{Var}(W(t)) = t ), this becomes ( text{Var}(Y) = sigma^2 t ).So, ( Y sim Nleft( left( alpha - frac{sigma^2}{2} right) t, sigma^2 t right) ).Now, the expectation ( E[e^Y] ) for a normal random variable ( Y ) with mean ( mu ) and variance ( sigma^2 ) is given by ( e^{mu + frac{sigma^2}{2}} ). This is a standard result from probability theory.Applying this to our case, ( E[e^Y] = e^{E[Y] + frac{text{Var}(Y)}{2}} ).Substituting the values we have:( E[e^Y] = e^{left( alpha - frac{sigma^2}{2} right) t + frac{sigma^2 t}{2}} ).Simplify the exponent:( left( alpha - frac{sigma^2}{2} right) t + frac{sigma^2 t}{2} = alpha t - frac{sigma^2 t}{2} + frac{sigma^2 t}{2} = alpha t ).So, ( E[e^Y] = e^{alpha t} ).Therefore, going back to ( E[X(t)] ):( E[X(t)] = X_0 e^{alpha t} ).Okay, so that's the expected value. It grows exponentially with rate ( alpha ), which makes sense because the drift term is ( alpha X(t) dt ).Now, moving on to the variance. The variance of ( X(t) ) is ( text{Var}(X(t)) = E[X(t)^2] - (E[X(t)])^2 ).We already have ( E[X(t)] = X_0 e^{alpha t} ), so ( (E[X(t)])^2 = X_0^2 e^{2 alpha t} ).So, we need to compute ( E[X(t)^2] ).From the solution of the SDE, ( X(t) = X_0 expleft( left( alpha - frac{sigma^2}{2} right) t + sigma W(t) right) ).Therefore, ( X(t)^2 = X_0^2 expleft( 2 left( alpha - frac{sigma^2}{2} right) t + 2 sigma W(t) right) ).So, ( E[X(t)^2] = X_0^2 Eleft[ expleft( 2 left( alpha - frac{sigma^2}{2} right) t + 2 sigma W(t) right) right] ).Again, let me denote ( Z = 2 left( alpha - frac{sigma^2}{2} right) t + 2 sigma W(t) ).Then, ( Z ) is a normal random variable with mean ( E[Z] = 2 left( alpha - frac{sigma^2}{2} right) t ) and variance ( text{Var}(Z) = (2 sigma)^2 text{Var}(W(t)) = 4 sigma^2 t ).Therefore, ( Z sim Nleft( 2 left( alpha - frac{sigma^2}{2} right) t, 4 sigma^2 t right) ).Using the same result as before, ( E[e^Z] = e^{E[Z] + frac{text{Var}(Z)}{2}} ).Substituting the values:( E[e^Z] = e^{2 left( alpha - frac{sigma^2}{2} right) t + frac{4 sigma^2 t}{2}} ).Simplify the exponent:First, expand ( 2 left( alpha - frac{sigma^2}{2} right) t ):= ( 2 alpha t - sigma^2 t ).Then, add ( frac{4 sigma^2 t}{2} = 2 sigma^2 t ).So, total exponent:( 2 alpha t - sigma^2 t + 2 sigma^2 t = 2 alpha t + sigma^2 t ).Therefore, ( E[e^Z] = e^{(2 alpha + sigma^2) t} ).Thus, ( E[X(t)^2] = X_0^2 e^{(2 alpha + sigma^2) t} ).Now, compute the variance:( text{Var}(X(t)) = E[X(t)^2] - (E[X(t)])^2 = X_0^2 e^{(2 alpha + sigma^2) t} - X_0^2 e^{2 alpha t} ).Factor out ( X_0^2 e^{2 alpha t} ):= ( X_0^2 e^{2 alpha t} left( e^{sigma^2 t} - 1 right) ).So, that's the variance.Let me recap:1. The expected value ( E[X(t)] = X_0 e^{alpha t} ).2. The variance ( text{Var}(X(t)) = X_0^2 e^{2 alpha t} (e^{sigma^2 t} - 1) ).I think that makes sense. The variance grows exponentially as well, but with a rate that includes both ( alpha ) and ( sigma ). The term ( e^{sigma^2 t} ) comes from the stochastic part, so as ( sigma ) increases, the variance increases more rapidly.Let me just double-check the steps.First, solving the SDE: yes, it's a geometric Brownian motion, and the solution is correct.Then, computing ( E[X(t)] ): used the fact that the expectation of the exponential of a normal variable is ( e^{mu + sigma^2 / 2} ). Applied it correctly, ended up with ( e^{alpha t} ).For the variance, computed ( E[X(t)^2] ) by recognizing that squaring the exponential leads to another exponential with doubled coefficients, then applied the same expectation formula, resulting in ( e^{(2 alpha + sigma^2) t} ). Then subtracted the square of the expectation, which is ( e^{2 alpha t} ), leading to the variance expression.Yes, that all seems consistent.I think that's it. So, the expected value is exponential growth with rate ( alpha ), and the variance is exponential growth with rate ( 2 alpha + sigma^2 ), minus 1, all multiplied by ( X_0^2 e^{2 alpha t} ).Final Answer1. The expected value is ( boxed{E[X(t)] = X_0 e^{alpha t}} ).2. The variance is ( boxed{text{Var}(X(t)) = X_0^2 e^{2 alpha t} left(e^{sigma^2 t} - 1right)} ).</think>