Show $\frac{1}{2}<\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n^p}<1$ for all positive number $p$.
112517
Proving bounds of a harmonic series
30407
Prove that $\sum_{n=1}^\infty\frac{n}{n^2+1}$ is divergent
132182
$\sum_{n=1}^\infty \sum_{m=1}^\infty\frac{1}{(m+n)^2}$
136444
Evaluating $\sum_{n=0}^\infty\frac{(-1)^n}{3n+1}$
134007
Borel-Cantelli-related exercise: Show that $\sum_{n=1}^{\infty} p_n < 1 \implies \prod_{n=1}^{\infty} (1-p_n) \geq 1- S$.
152698
Evaluation of $\sum_{n=0}^{\infty}\frac{1}{(n^4+n^2+1)n!}$
156367
How to prove that $\sum\limits_{n=1}^{\infty}\frac{1}{n(n+1)(n+2)...(n+k)} = \frac{1}{kk!}$ for every $k\geqslant1$
108313
For which values of $p$ does the serie $\sum\limits_{n=2}^\infty\frac{1}{n^p\ln(n)}$ converge?
79415
The series $\sum_{n=2}^{\infty }\frac{1}{\log (n!)}$
67752
Show that $\sum_{n = 1}^{+\infty} \frac{n}{2^n} = 2$
94792
Convergence of a series $\sum\limits_{n=1}^\infty\left(\frac{a_n}{n^p}\right)^\frac{1}{2}$
16143
How can I prove that $\sum_{n=1}^\infty \frac{1}{n(n+1)} = 1$?
125701
Prove that $-2\log(2) = -2 + \sum_{n=1}^{\infty}\frac{1}{n(2n+1)}$
65249
An interesting exercise about converging positive series, involving $\sum_{n\geq 1}a_n^{\frac{n-1}{n}}$
116723
Evaluating the sum $\sum_{n=1}^{\infty}\dfrac{(-1)^{n}}{n^{2}}$
117863
Series $\sum_{n=1}^\infty \frac{n^n}{(n!)^2}$
63380
Show that the series $\sum_{n=2}^{\infty} \frac{1}{n\, (\log n)^s}$ converges for $s >1$ and diverges for $s=1$