Newbetuts

Find $\sum_{n=1}^{\infty}$ $\frac{n^{2}}{\left(n+1\right)\left(n+2\right)\left(n+3\right)\left(n+4\right)}$

Cauchy-Ramanujan Formula $ \displaystyle \sum_{\stackrel{m \in \mathbb{Z}}{m \neq 0}} \frac{\coth m \pi}{m^{4p+3}} $

Show $ f(x) = \sum_{n=1}^{\infty} \frac{nx}{n^3 + x^3}$ ,$\ g(x) = \sum_{n=1}^{\infty} \frac{x^4n}{(n^3 + x^3)^2}$ are bounded on $[0, \infty)$.

Convergence or divergence of $\sum_{k=1}^{\infty} \left(1-\cos\frac{1}{k}\right)$ [duplicate]

How long does a sequence need to be to be guaranteed to have a monotonic subsequence length k?

Ways of showing $\sum_\limits{n=1}^{\infty}\ln(1+1/n)$ to be divergent

why is $\sum_{n=1}^{\infty} \frac{1}{n} \sin\left( \frac{x^n}{\sqrt n} \right)$ uniformly convergent? [duplicate]

Is this a valid argument for proving that a sum of reciprocals is irrational?

Proof of a Combinatorial Abel Identity

Compute $\sum\limits_{n=1}^\infty\frac{1}{(n(n+1))^p}$ where $p\geq 1$

Proving that $\sum_{k=0}^{2n} {2k \choose k } {2n \choose k}\left( \frac{-1}{2} \right)^k=4^{-n}~{2n \choose n}.$

A series for $\log (a) \log (b)$ in terms of hypergeometric function

Do non-$\ell^2$ sequences have an $\ell^2$ functional that takes them to infinity? [duplicate]

Series $\sum \frac{\sin(n)}{n} \cdot \left(1+\cdots +\frac{1}{n}\right)$ convergence question

What is known about the minimal number $f(n)$ of geometric progressions needed to cover $\{1,2,\ldots,n\}$, as a function of $n$?

How do we know Taylor's Series works with complex numbers?

New posts in sequences-and-series

Find $\sum_{n=1}^{\infty}$ $\frac{n^{2}}{\left(n+1\right)\left(n+2\right)\left(n+3\right)\left(n+4\right)}$

How to decide about the convergence of $\sum(n\log n\log\log n)^{-1}$?

Show the sequence converges to M

How to evaluate $ \sum\limits_{n=1}^{\infty} \left( \frac{H_{n}}{(n+1)^2.2^n} \right)$

$\lim_{n\to\infty} \frac{1}{\log(n)}\sum _{k=1}^n \frac{\cos (\sin (2 \pi \log (k)))}{k}$

Cauchy-Ramanujan Formula $ \displaystyle \sum_{\stackrel{m \in \mathbb{Z}}{m \neq 0}} \frac{\coth m \pi}{m^{4p+3}} $

Show $ f(x) = \sum_{n=1}^{\infty} \frac{nx}{n^3 + x^3}$ ,$\ g(x) = \sum_{n=1}^{\infty} \frac{x^4n}{(n^3 + x^3)^2}$ are bounded on $[0, \infty)$.

Convergence or divergence of $\sum_{k=1}^{\infty} \left(1-\cos\frac{1}{k}\right)$ [duplicate]

How long does a sequence need to be to be guaranteed to have a monotonic subsequence length k?

Ways of showing $\sum_\limits{n=1}^{\infty}\ln(1+1/n)$ to be divergent

why is $\sum_{n=1}^{\infty} \frac{1}{n} \sin\left( \frac{x^n}{\sqrt n} \right)$ uniformly convergent? [duplicate]

Is this a valid argument for proving that a sum of reciprocals is irrational?

Proof of a Combinatorial Abel Identity

Compute $\sum\limits_{n=1}^\infty\frac{1}{(n(n+1))^p}$ where $p\geq 1$

Proving that $\sum_{k=0}^{2n} {2k \choose k } {2n \choose k}\left( \frac{-1}{2} \right)^k=4^{-n}~{2n \choose n}.$

A series for $\log (a) \log (b)$ in terms of hypergeometric function

Do non-$\ell^2$ sequences have an $\ell^2$ functional that takes them to infinity? [duplicate]

Series $\sum \frac{\sin(n)}{n} \cdot \left(1+\cdots +\frac{1}{n}\right)$ convergence question

What is known about the minimal number $f(n)$ of geometric progressions needed to cover $\{1,2,\ldots,n\}$, as a function of $n$?

How do we know Taylor's Series works with complex numbers?