New posts in Sequences And Series

Confused on what a series means.

Closed form of the sum $\sum_{n=1}^{\infty}\frac{H_n}{n^x}$

Given the sequence $a_1=1$,$a_{n+1}=1+\frac{n}{a_n}$, does the sequence $a_n+n-a_n^2$ converges?

Is $\limsup_{t} \frac{a_t}{ h(t+1) - h(t) }$ finite when $0 \leq \sum_{i=t}^\infty a_i \leq h(t) = t^{-\alpha}$?

Prove that a summation is convergent

Is this a correct way to show that $\sum_{n \geq 0} \frac{n^3}{n!}=5e$

Show the convergence of $\sum_{n=0}^{\infty}\frac{1}{2}\cdot \frac{(2n)!}{n!(n+1)!}\cdot \left(\frac{1}{4}\right)^n$

Show that $\lim\limits_{n \to +\infty}(\sin(\frac{1}{n^2})+\sin(\frac{2}{n^2})+\cdots+\sin(\frac{n}{n^2})) = \frac{1}{2}$

the limit of $\frac{1}{\sqrt{n}}(1+\frac{2}{1+\sqrt2}+\frac{3}{1+\sqrt2+\sqrt3}+\dots+\frac{n}{1+\sqrt2+\sqrt3+\dots+\sqrt n})$ as $n\to\infty$

Limit of $a_{n+1} = a_{n}(1+1/n^2)$