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

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:.

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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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Integration by parts yields Dn=12n+1∫π/20f′(x)cos(2n+1)x dx=O(1/n). .... For θ not an integer multiple of 2π, 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.

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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.

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Harmonic series (mathematics) - Wikipedia

History[edit]. The fact that the harmonic series diverges was first proven in the 14th century by Nicole Oresme, but this achievement fell into obscurity. Proofs were given in the 17th century by Pietro Mengoli, Johann Bernoulli, ... If the worm travels 1 centimeter per minute and the band stretches 1 meter per minute, will the ...

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