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Let $\langle \cdot, \cdot \rangle$ denote the usual inner product of $\mathbb{R}^m$. Evaluate the integral $$\mathcal{M} = \int \limits_{\mathbb{R}^m} \exp \left( - ( \langle x, \mathcal{S} ^{-1} x \rangle )^a \right) \, {\rm d}x$$ where $\mathcal{S}$ is a positive symmetric $m \times m$ matrix and ...

Let $\mathcal{H}_n$ denote the $n$-th harmonic number. Prove that forall $|x|<1$ it holds that

\[\sum_{n=1}^{\infty} \frac{(-1)^{n-1} \mathcal{H}_{2n} x^{2n+1}}{2n+1} = \frac{\arctan x \log (1+x^2)}{2}\]

Grigorios Kostakos wrote:\(\psi\big(n+\frac{1}{2}\big)-\psi(n)\),

Well,

$$\psi^{(0)}\left( n + \frac{1}{2} \right) = -\gamma +2 \ln 2 +2 \sum_{k=1}^{n} \frac{1}{2k-1}$$

and

$$\psi^{(0)}(n) = - \gamma +\sum_{k=1}^{n-1} \frac{1}{k}$$

I suppose you don't want Gauss digamma theorem to be used, e? , but rather a more elegant one.

Here is something that came up today while evaluating something else.

Prove that

$$\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \frac{1}{n^6 \left ( m^2+n^2 \right )} = \frac{13 \pi^8}{113400}$$

Evaluate the series

$$\mathcal{S} = \sum_{n=1}^{\infty} \frac{ \cos \left( \frac{n \pi}{12} \right)}{n 3^n}$$

The file is updated! Check first post for a direct download.

We have seen (somewhere in the past) the Fourier series of $\cos x$ as well as $\sin ax$. How about the Fourier series of $\sec x$?

Let $x \in \left( - \frac{\pi}{4} , \frac{\pi}{4} \right)$. Expand $\sec x$ in a Fourier series.

Let $\alpha \in \mathbb{R} \setminus \mathbb{Z}$ and let us denote with $\lfloor \cdot \rfloor$ the floor function. Prove that the series

$$\mathcal{S}= \sum_{n=1}^{\infty} \left(\alpha-\frac{\lfloor n\alpha \rfloor}{n}\right)$$

diverges.

The following series is an interesting one because of its slow convergence which you are asked to show! It was a question at École Polytechnique.

Prove that the series

$$\mathcal{S} = \sum_{n=1}^{\infty} \frac{(-1)^{n-1} |\sin n|}{n}$$

converges but not absolutely.