Intereting Posts

Proof that $\mathbb{Z}\left$ is a PID
Infiniteness of twin prime powers
Prove that $(p+q)^m \leq p^m+q^m$
Polyhedra from number fields
Prove: $\cot x=\sin x\,\sin\left(\frac{\pi}{2}-x\right)+\cos^2x\,\cot x$
Induction with floor, ceiling $n\le2^k\implies a_n\le3\cdot k2^k+4\cdot2^k-1$ for $a_n=a_{\lfloor\frac{n}2\rfloor}+a_{\lceil\frac{n}2\rceil}+3n+1$
Double Integration with change of variables
Good book on integral calculus (improper integrals, integrals with parameters, special functions)
Use Integration by Parts to prove that $\int x^{n}\ln{x}\ dx=\frac{x^{n+1}}{(n+1)^{2}}\left+c$
Weak convergence and integrals
Prime candidates of the form $n^{(n^n)}+n^n+1$?
Definition of characteristic polynomial
Test for the convergence of the sequence $S_n =\frac1n \left(1 + \frac{1}{2} + \frac{1}{3} + \cdots+ \frac{1}{n}\right)$
How is $e$ the only number $n$ for which $n^x > x^n$ is satisfied for all values of $x$?
Using inequalities and limits

I wanted to evaluate the sum:

$$ \sum_{n \ge 2} \left(\zeta(n) – 1\right) $$

I rewrote this as:

- Closed form of $\lim\limits_{n\to\infty}\left(\int_0^{n}\frac{{\rm d}k}{\sqrt{k}}-\sum_{k=1}^n\frac1{\sqrt k}\right)$
- Two Dirichlet's series related to the Divisor Summatory Function and to the Riemann's zeta-function, $\zeta(s)$
- Proof that $\sum\limits_{k=1}^\infty\frac{a_1a_2\cdots a_{k-1}}{(x+a_1)\cdots(x+a_k)}=\frac{1}{x}$ regarding $\zeta(3)$ and Apéry's proof
- Apéry's constant ($\zeta(3)$) value
- Riemann zeta function and Bernoulli function
- Proof of Riemann Hypothesis

$$ \sum_{n\ge 2} \sum_{k\ge 2} \frac{1}{n^k} $$

I tried exploiting the symmetry but that didn’t seem to help. I know from numerical calculation that the answer is 1.

- closed form for a double sum
- Elementary derivation of certian identites related to the Riemannian Zeta function and the Euler-Mascheroni Constant
- Prove the following identity ${(\sum_1^na_jb_j)}^2 = {(\sum_1^na^2_j)}{(\sum_1^nb^2_j)}^2-\sum_1^n\sum_1^n{(a_kb_j-a_jb_k)}^2$
- Is $\sqrt1+\sqrt2+\dots+\sqrt n$ ever an integer?
- How does $\zeta(1 - s)$ become $(-1/s + \cdots)$?
- Certain step in the induction proof $\sum\limits_{i=0}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$ unclear
- Proof that $\sum_{i=1}^n{1} = n$ for all $n \in \Bbb Z^+$
- Help with proof using induction: $1 + \frac{1}{4} + \frac{1}{9}+\cdots+\frac{1}{n^2}\leq 2-\frac{1}{n}$
- Proving a function is continuous and periodic
- Evaluating even binomial coefficients

As Shalop points out, the inner sum is the geometric series $\frac{1}{n^2}+\frac{1}{n^3}+\cdots=\frac{1/n^2}{1-1/n}=\frac{1}{n(n-1)}$ which admits the partial fraction decomposition $\frac{1}{n-1}-\frac{1}{n}$. Thus your sum simplifies to the telescoping series $(1-\frac{1}{2})+(\frac{1}{2}-\frac{1}{3})+(\frac{1}{3}-\frac{1}{4})+\cdots$ which you should know how to do.

- Using trig substitution to evaluate $\int \frac{dt}{( t^2 + 9)^2}$
- Is $ (A × B) ∪ (C × D) = (A ∪ C) × (B ∪ D)$ true for all sets $A, B, C$ and $D$?
- $P(X^2+Y^2<1)$ of two independent n(0,1) random variables
- Convergence of $x_n = \cos (x_{n-1})$
- Contraction of non-zero prime ideals in the ring of algebraic integers
- How do we take second order of total differential?
- Show that $\lim_{n \rightarrow \infty}\frac{a_n}{b_n}=1 \implies \lim_{n \rightarrow \infty}{(a_n-b_n)}=0$
- Omitting the hypotheses of finiteness of the measure in Egorov theorem
- What exactly determines the block-sizes for Jordan forms?
- Is there a problem for which it is known that the only solution is “iterative”?
- Triangle problem related to finding an area
- Show that the $\Delta$-complex obtained from $\Delta^3$ by performing edge identifications deformation retracts onto a Klein bottle.
- How to geometrically show that there are $3$ $D_4$ subgroups in $S_4$?
- Does $\mathbb Q(\sqrt{-2})$ contain a square root of $-1$?
- Terminology re: continuity of discrete $a\sin(t)$