35 questions linked to/from Nice proofs of $\zeta(4) = \frac{\pi^4}{90}$?
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### Potentially Useful Question [duplicate]

I have been solving problems using a "Potentially Helpful Formulas" sheet from my esteemed math professor. i want to solve for: $\sum_{n=1}^{\infty} \dfrac{1}{n^4} =$ ? On my formula sheet i have: ...
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### $\sum_{n=1}^\infty \frac{1}{n^4}=\frac{\pi^4}{90}$ [duplicate]

How we can do this sum? $$\sum_{n=1}^\infty \frac{1}{n^4}=\frac{\pi^4}{90}$$ I know that we could possibly use a Fourier series decomposition however I don't know what function to start with. I ...
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I was doing my homework assignment and I did this question correctly. However, I'm interested in knowing the reason behind the logic that how $1/1^4 + 1/2^4 + 1/3^4$ .... to infinity evaluates to $pi^... 0answers 32 views ### Infinite Sum Convergence [duplicate] What value does the infinite sum$\sum_{k=1}^{\infty} \frac{1}{k^4} $converge to? I know that$\sum_{k=1}^{\infty} \frac{1}{k^2} $converges to$\frac{\pi^2}{6}$but I simply do not have a clue as to ... 43answers 96k views ### Different methods to compute$\sum\limits_{k=1}^\infty \frac{1}{k^2}$(Basel problem) As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem) $$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler ... 5answers 4k views ### Computing$\zeta(6)=\sum\limits_{k=1}^\infty \frac1{k^6}$with Fourier series. Let$ f$be a function such that$ f\in C_{2\pi}^{0}(\mathbb{R},\mathbb{R}) $(f is$2\pi$-periodic) such that$ \forall x \in [0,\pi]$: $$f(x)=x(\pi-x)$$ Computing the Fourier series of$f$and ... 3answers 1k views ### interesting square of log sin integral I ran across this challenging log sin integral and am wondering what may be a good approach. $$\int_{0}^{\frac{\pi}{2}}x^{2}\ln^{2}(2\cos(x))dx=\frac{11{{\pi}^{5}}}{1440}$$ This looks like it ... 5answers 655 views ### The other ways to calculate$\int_0^1\frac{\ln(1-x^2)}{x}dx\$
Prove that $$\int_0^1\frac{\ln(1-x^2)}{x}dx=-\frac{\pi^2}{12}$$ without using series expansion. An easy way to calculate the above integral is using series expansion. Here is an example \begin{...