I want to show that $\displaystyle \int_{-\infty}^{\infty} \frac{\sinh ax}{\sinh \pi z} \cos bx \ dx = \frac{\sin a}{\cos a + \cosh b} \ ( -\pi < a < \pi) . $ So I let $f \displaystyle (z) = e^{ibz} \frac{\sinh az}{\sinh \pi z} $ and integrated around a rectangular contour with vertices at $R, R...

Find the value of the limit: $$\lim_{n\to\infty} \sum_{k=0}^n \frac{{k!}^{2} {2}^{k}}{(2k+1)!}$$ I'm trying to find out if this limit can be computed only by using high school knowledge for solving limits. Thanks.

This is a question about pointwise convergence of a Fourier transform of functions of the form $f: \mathbb{R}^N \to \mathbb{R}$, which is potentially a $N$-dimensional generalization to pointwise convergence of $1$ dimensional Fourier transform. This question arose when I am trying to generalize ...

I'm sure you can do this easily, but I'm looking for an easy way that only uses series manipulation. Is that possible? $$\sum_{n=1}^{\infty} \left(\psi^{(0)}\left(\frac{1+n}{2}\right)-\psi^{(0)}\left(\frac{n}{2}\right)-\frac{1}{n}\right)$$

$$\sum_{n=1}^{\infty} \left(\psi^{(0)}\left(\frac{1+n}{2}\right)-\psi^{(0)}\left(\frac{n}{2}\right)-\frac{1}{n}\right)^{2014}=$$ (-155015596184364440945084299975759662568630079642309524338450663434751218053297478865897807539291708732320332850003195271367671359884906941357534021938515281380329185...

I'm sure you can do this easily, but I'm looking for an easy way that only uses series manipulation. Is that possible? $$\sum_{n=1}^{\infty} \left(\psi^{(0)}\left(\frac{1+n}{2}\right)-\psi^{(0)}\left(\frac{n}{2}\right)-\frac{1}{n}\right)$$ where $\psi^{(0)}(x)$ is digamma function Here is a su...

From where can I practice questions related to the following topics: Arrow's Impossibility Theorem Nash's Axiomatic Bargaining. Harsanyi's Axiomatic characterization of Utilitarianism. ?

Let us take as an example this question. Consider whether you think it should be considered off-topic (for being purely mathematical in nature). Should one's opinion on whether a mathematical question be closed be based on what one finds interesting, the specific written rules of this site, a...

I'm sure you can do this easily, but I'm looking for an easy way that only uses series manipulation. Is that possible? $$\sum_{n=1}^{\infty} \left(\psi^{(0)}\left(\frac{1+n}{2}\right)-\psi^{(0)}\left(\frac{n}{2}\right)-\frac{1}{n}\right)$$ where $\psi^{(0)}(x)$ is digamma function Here is a su...