Intereting Posts

Multiplicative Inverse Element in $\mathbb{Q}{2}]$
Prove that for any $1 < p < ∞$ there exists a function $f ∈ L_p(μ)$ such that $f \notin L_q(μ)$ for any $q > p.$
minimizing sum of distances
Proof that if $\phi \in \mathbb{R}^X$ is continuous, then $\{ x \mid \phi(x) \geq \alpha \}$ is closed.
If a polynomial divides another in a polynomial ring, will the division also occur in a subring?
Factorial of infinity
Solving Differential Functional Equation $f(2x)=2f(x)f'(x)$
Covariant derivative versus exterior derivative
Approximation of integrable function by continuous in norm of $L_2$
Help with proof using induction: $1 + \frac{1}{4} + \frac{1}{9}+\cdots+\frac{1}{n^2}\leq 2-\frac{1}{n}$
$4$ or more type $2$ implies $3$ or less type $1$
Prove that the congruence $x^2 \equiv a \mod m$ has a solution if and only if for each prime $p$ dividing $m,$ one of the following conditions holds
How to exhibit models of set theory
Whenever Pell's equation proof is solvable, it has infinitely many solutions
How to determine the Galois group of irreducible polynomials of degree $3,4,5$

Tried using $b_n = \frac1{n^n + 1}$ with limit test which indicated that both either converge or diverge but getting stuck on how to show that one actually does converge.

- For bounded sequences, does convergence of the Abel means imply that for the Cesàro means?
- Given some power series, how do I make one centered at a different point?
- What is the function given by $\sum_{n=0}^\infty \binom{b+2n}{b+n} x^n$, where $b\ge 0$, $|x| <1$
- Grandi's series contradiction
- Summation of natural number set with power of $m$
- A limit involves series and factorials

- Need help finding a closed form for complicated sum
- Divergence of $\sum\limits_n1/\max(a_n,b_n)$
- Evaluation of $\prod_{n=1}^\infty e\left(\frac{n}{n+1}\right)^{n}\sqrt{\frac{n}{n+1}}$
- Techniques to prove that there is only one square in a given sequence
- Is the sum of all natural numbers $-\frac{1}{12}$?
- A sum for stirling numbers Pi, e.
- Evaluating a sum involving binomial coefficient in denominator
- Proof that a sequence converges to a finite limit iff lim inf equals lim sup
- Proving that $\left(\frac{\pi}{2}\right)^{2}=1+\sum_{k=1}^{\infty}\frac{(2k-1)\zeta(2k)}{2^{2k-1}}$.
- Convergence of a product series with one divergent factor

Use that $\sum\limits_{k=1}^{\infty} a_k $ converges if and only if $\sum\limits_{k=j}^{\infty} a_k $ converges for a $j \in \mathbb N$

Edit: Here is the full solution:

$\sum\limits_{n=1}^{\infty} \frac{1}{n^n}<\infty \iff \sum\limits_{n=2}^{\infty}

\frac{1}{n^n}<\infty$

and $ \sum\limits_{n=2}^{\infty} \frac{1}{n^n}=\frac{1}{2^2}+\frac{1}{3^3}+\frac{1}{4^4}+…. \leq \frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+….=\sum\limits_{n=2}^{\infty} \frac{1}{n^2}< \infty$

You can conclude if the series converges or not,using the ratio test.

It is like that:

$a_n=\frac{1}{n^n}$

$$\frac{a_{n+1}}{a_n}=\frac{\frac{1}{(n+1)^{n+1}}}{\frac{1}{n^n}}=\frac{n^n}{(n+1)^{n+1}} \to 0<1$$

So,the series $\sum_{n=1}^{\infty} \frac{1}{n^n}$ converges.

- Entropy of a natural number
- How can I find equivalent Euler angles?
- How can the Laplace transform be used to solve piecewise functions?
- In $\ell^1$ but not in $\ell^2$?
- How to calculate the integral of $x^x$ between $0$ and $1$ using series?
- Alternative “functorial” proof of Nielsen-Schreier?
- Find the modulo between two large number
- Find the domain of $x^{2/3}$
- Spectrum of shift-operator
- A simple way to obtain $\prod_{p\in\mathbb{P}}\frac{1}{1-p^{-s}}=\sum_{n=1}^{\infty}\frac{1}{n^s}$
- How many copies of $C_4$ are there in $K_n$
- About the limit of a uniformly converging function sequence
- Is a basis for the Lie algebra of a Lie group also a set of infinitesimal generators for the Lie group?
- Find all Integral solutions to $x+y+z=3$, $x^3+y^3+z^3=3$.
- Minimum of $ f(\alpha) = \left(1+\frac{1}{\sin^{n}\alpha}\right)\cdot \left(1+\frac{1}{\cos^{n}\alpha}\right)$