How to prove that $1^r + 2^r +... + n^r = a_1 n^1 + a_2 n^2 +... + a_{r+1} n^{r+1}$?
35525
Methods to compute $\sum_{k=1}^nk^p$ without Faulhaber's formula
114173
Explanation of the formulas for sums $\sum nr^n$ and $\sum n^2 r^n$
158004
A formula for $1^4+2^4+...+n^4$
53918
whats the proof for $\lim_{x → 0} [(a_1^x + a_2^x + .....+ a_n^x)/n]^{1/x} = (a_1.a_2....a_n)^{1/n}$
48503
Proving : $ \bigl(1+\frac{1}{n+1}\bigr)^{n+1} \gt (1+\frac{1}{n})^{n} $
77076
Proof that $(x-1)^2$ divides a $nx^{n+1} – (n+1)x^{n}+1$
26741
How to prove for $s<1,|a+b|^s\le|a|^s+|b|^s$
86054
$r+r^{-1}$ integral implies $r^n+r^{-n}$ integral
110996
Sum of the form $r+r^2+r^4+\dots+r^{2^k} = \sum_{i=1}^k r^{2^i}$
102464
Prove $R < ( 1 + (\sum |a_{i}|^{p} )^{q/p})^{1/q}$ for absolute value of polynomial zeros
60036
How to prove this limit exists: $a_{n+2}=\sqrt{a_{n+1}}+\sqrt{a_{n}}$
89175
Proof of $(a+b)^{n+1}$
109532
Proving $5^n \equiv 1 \pmod {2^r}$ when $n=2^{r-2}$
96097
Let $k$ be an odd positive integer. Prove that $(1+2+3...+n)|(1^k+2^k+3^k+...+n^k)$ for all positive integers $n$.
89728
Prove that $1<\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{3n+1}$
12020
Prove $\sqrt{n+1}-\sqrt{n}\lt \frac{1}{2\sqrt{n}}$
78144
How to prove the inequality $a_1^{a_1}a_2^{a_2}\cdots a_n^{a_n} > \left(\frac{a_1+\dots+a_n}{n}\right)^{a_1+\dots+a_n}$?