New posts in Sequences And Series

Regarding evaluation of the limit of the sequence $\Bigl(\frac{1}{n}\Bigr)^n+\Bigl(\frac{2}{n}\Bigr)^n+ \cdots \Bigl(\frac{n}{n}\Bigr)^n$

Is the sequence $x_n=\dfrac{1}{\sqrt{n}}\left(1+\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{3}}+\ldots+\dfrac{1}{\sqrt{n}}\right)$ monotone?

Evaluate $\sum_{n,k} \binom{n}{k}^{-1} $

A closed form for $ \int_{0}^{\frac{\pi}{2}}{\left(\frac{\sin{\left(nx\right)}}{\sin{x}}\right)^{2p}\,\mathrm{d}x} $?

Does $\lim_{N\rightarrow \infty} \sum_{n = 1}^{N} \frac{1}{(N+1) \ln (N+1) - n \ln n } = 1$?

Find value of $p$ to make the series $\sum\limits_{n=1}^\infty\left(\dfrac1{n^p}\sum\limits_{k=1}^nk^{3/2}\right)$ converge

Why is the sequential criterion true in metric spaces? When does it fail in general?

How can we prove that $\gcd((n^4) + (n+1)^4 , (n+1)^4 + (n+2)^4) = 1$?

Erase every other number from $1, 2, 3, ... 2000$ until only one number left

Proving that $u_{n}\in\mathbb{Z}\forall n\ge 0$, such that $\det \begin{bmatrix}u_{n} & u_{n+1} \\ u_{n+2}&u_{n+3}\end{bmatrix}=n!\forall n\ge 0$