Mahesh Godavarti
3

Good question.

Here is an interesting way to think about it.

\sum_{n=1}^{\infty} \frac{1}{n^2} = \int_1^{\infty} \frac{1}{\lfloor x \rfloor^2} dx .

\int_1^{\infty} \frac{1}{x^2} dx is the area under \frac{1}{x^2} in $[1, \infty)$, whereas \sum_{n=1}^{\infty} \frac{1}{n^2} is the area under \frac{1}{\lfloor x \rfloor^2} in $[1, \infty)$.

This is illustrated in the following picture (where the green curve is 1/x^2 , the solid red curve is 1/n^2 and the dotted red curve is 1/(n+1)^2 .

One can see from the picture that \int_1^{\infty} \frac{1}{x^2} dx < \sum_{n=2}^\infty \frac{1}{n^2} + 1/2 (\sum_{n=1}^\infty \frac{1}{n^2} - \sum_{n=2}^\infty \frac{1}{n^2}) = \sum_{n=1}^{\infty} \frac{1}{n^2} - 1/2 \approx 1.145

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calculus - Nice proofs of $\zeta(4) = \frac{\pi^4}{90}$? - Mathematics ...

I know some nice ways to prove that ζ(2)=∑∞n=11n2=π2/6. ... If we substitute π for x in the Fourier trigonometric series expansion of f(x)=x4, with ..... It is not hard to guess that p(z) is same as isin(z)×sin(zi)z2=(1−z23!+z45! .... Consider the contour integral ∮Cπcotπzz4 dz ..... The second sum in the left member is equal to:.

For more information, see calculus - Nice proofs of $\zeta(4) = \frac{\pi^4}{90}$? - Mathematics ...

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Pi Formulas -- from Wolfram MathWorld

pi is intimately related to the properties of circles and spheres. For a circle of radius r ... terms is approx (3/4)^k . An infinite sum series to Abraham Sharp (ca. 1717) is ... 105-106). The coefficients can be found from the integral ... At the cost of a square root, Gosper has noted that x=1/2 gives 2 ... 1999) and is equivalent to ...

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Gamma Function -- from Wolfram MathWorld

8). It is analytic everywhere except at z=0 , -1 , -2 , ..., and the residue at z=-k is ... The gamma function is implemented in the Wolfram Language as Gamma[z]. ... Integrating equation (3) by parts for a real argument, it can be seen that ... The gamma function can also be defined by an infinite product form (Weierstrass form ) ...

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calculus - Evaluating the integral $\int_0^\infty \frac{\sin x} x \ dx ... Integration by parts yields Dn=12n+1π/20f′(x)cos(2n+1)x dx=O(1/n). .... For θ not an integer multiple of , we have ∑einθn=−log(1−eiθ). .... Here is a sketch of another elementary solution based on a proof in Bromwich's Theory of Infinite Series. ...... to deduce it is exactly 1sinx, so the RHS of (1) simply equals π. For more information, see calculus - Evaluating the integral$\int_0^\infty \frac{\sin x} x \ dx ...

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differential equations - What is wrong with my approach to solving a ...

Jan 18, 2019 ... Infinity -> 20 // Activate, x] // Evaluate, {x, 0, Pi}, {t, 0, 1}] ..... your Exp[-n^2 t] only damps out part of the sum while the sum damps out very rapidly in the separate variable case. .... Again, each side must be equal to a constant. .... c3*Integrate[ Cos[n*x]^2, {x, 0, Pi}] == (1/Pi)*Integrate[x^2*Cos[n*x], {x, 0, Pi}] ...

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Basel problem - Wikipedia

Squaring the circle; Basel problem; Six nines in π · Other topics related to π · v · t · e. The Basel problem is a problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734 and read on 5 December 1735 in The ... The sum of the series is approximately equal to 1.644934.