Conjecture: $\sum\limits_{n\geq0}\left(\frac12\right)^n\prod\limits_{k=1}^{n}\frac{2n-2k+1}{2n-2k+2}=\sqrt2$
59253
Summing the power series $\sum\limits_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{2n+1}\prod\limits_{k=1}^n\frac{2k-1}{2k} $
54286
Let $p$ a prime number, then prove that : $\sum \limits_{k=0}^{p} \binom {p}{k} \binom{p+k}{k} \equiv 2^p +1 \pmod{p^2}$
30672
Show that $\sum\limits_{k=0}^n\binom{2n}{2k}^{\!2}-\sum\limits_{k=0}^{n-1}\binom{2n}{2k+1}^{\!2}=(-1)^n\binom{2n}{n}$
36083
If $k > 1$, then $\frac1{(k - 1)^2}-\frac1{(k + 1)^2}=\frac{4k}{(k^2 - 1)^2}$, hence simplify $\sum\limits_{k = 2}^n\frac k{(k^2 - 1)^2}$
123955
How to compute $\sum\limits_{k=0}^n (-1)^k{2n-k\choose k}$?
68858
Prove that $\prod_{n=2}^∞ \left( 1 - \frac{1}{n^4} \right) = \frac{e^π - e^{-π}}{8π}$
71736
I found this odd relationship, $x^2 = \sum_\limits{k = 0}^{x-1} (2k + 1)$.
78008
A conjectured closed form of $\int\limits_0^\infty\frac{x-1}{\sqrt{2^x-1}\ \ln\left(2^x-1\right)}dx$
64223
Infinite Series $\sum\limits_{n=1}^\infty\left(\frac{H_n}n\right)^2$
4414
Computig the series $\sum\limits_{n=2}^\infty \ln\left(1-\frac{1}{n^2}\right)$
56365
Evaluate $\sum\limits_{k=1}^{n} \frac{k}{2^k}$
113891
The limit of a sum of the form $a_0 \sqrt n + a_1 \sqrt {n + 1} +\cdots + a_k \sqrt {n + k}$
103843
Infinite Series $\sum\limits_{n=1}^{\infty}\frac{1}{\prod\limits_{k=1}^{m}(n+k)}$
64501
Homework: limit of $\frac{1}{n}\sqrt[n]{\prod_{k=1}^n(n+k)}$ as $n\to\infty$
20317
Proving $\sum_{k=1}^{n}{(-1)^{k+1} {{n}\choose{k}}\frac{1}{k}=H_n}$
64581
How to find the sum $\sum\limits_{k=1}^n (k^2+k+1)k!$?
155005
How to show $\sum_{k=n}^\infty{\frac{1}{k!}} \leq \frac{2}{n!}$