Intereting Posts

Elegant solution to $\lim\limits_{n \to \infty}{{a} – 1)]}$
Prove that if LCMs are equal, then the numbers are equal too.
Prove that $\mathbb Z_{m}\times\mathbb Z_{n} \cong \mathbb Z_{mn}$ implies $\gcd(m,n)=1$.
How prove this $(abc)^4+abc(a^3c^2+b^3a^2+c^3b^2)\le 4$
$\subset$ vs $\subseteq$ when *not* referring to strict inclusion
Does there exist such an invertible matrix?
Why is the minimum size of a generating set for a finite group at most $\log_2 n$?
Determine the PDF of Z = XY when the joint pdf of X and Y is given.
Fiber bundle is compact if base and fiber are
Groups of homeomorphisms of the real line
Bounded Variation $+$ Intermediate Value Theorem implies Continuous
how many element of order 2 and 5 are there
Is every quotient of a finite abelian group $G$ isomorphic to some subgroup of $G$?
Does TG prove that ZFC2 has a model?
An application of J.-L. Lion's Lemma

I wanted to evaluate the sum:

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

I rewrote this as:

- Can the Basel problem be solved by Leibniz today?
- Rational Roots of Riemann's $\zeta$ Function
- Computing $\zeta(6)=\sum\limits_{k=1}^\infty \frac1{k^6}$ with Fourier series.
- Asymptotics for zeta zeros?
- Importance of the zero free region of Riemann zeta function
- Books about the 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.

- What consistent rules can we use to compute sums like 1 + 2 + 3 + …?
- Finding the Moment Generating function of a Binomial Distribution
- Riemann's thinking on symmetrizing the zeta functional equation
- complicated derivative with nested summations
- Showing that $\sum_{i=1}^n \frac{1}{i} \geq \log{n}$
- Closed form for $\sum_{n=1}^{\infty}\frac{1}{\sinh^2\!\pi n}$ conjectured
- Showing $1+2+\cdots+n=\frac{n(n+1)}{2}$ by induction (stuck on inductive step)
- Proving that $\pi(2x) < 2 \pi(x) $
- Prime Number Theorem and the Riemann Zeta Function
- Real and imaginary part of Gamma function

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.

- Find the subgroups of A4
- $a,b,c,d\ne 0$ are roots (of $x$) to the equation $ x^4 + ax^3 + bx^2 + cx + d = 0 $
- Showing $\sum_{i = 1}^n\frac1{i(i+1)} = 1-\frac1{n+1}$ without induction?
- What can we learn about a group by studying its monoid of subsets?
- filling an occluded plane with the smallest number of rectangles
- A limit problem $\lim\limits_{x \to 0}\frac{x\sin(\sin x) – \sin^{2}x}{x^{6}}$
- Integration of forms and integration on a measure space
- Facebook Question (Data Science)
- A result of Erdős on increasing multiplicative functions
- The length of an interval covered by an infinite family of open intervals
- abel summable implies convergence
- Difference between Euclidean Space and $\mathbb{R}^n$ other than origin?
- (Counting problem) very interesting Modular N algebraic eqs – for combinatorics-permutation experts
- Find $x$: $5\cos(x) = 2\sec(x)-3$
- compute one improper integral involving arctangent