Intereting Posts

Examples of non symmetric distances
Quadratic Recurrence : $f(n) = f(n-1) + f(n-2) + f(n-1) f(n-2)$ Solution? How?
Solve $\frac{\cos x}{1+\sin x} + \frac{1+\sin x}{\cos x} = 2$
Duality of $L^p$ and $L^q$
Formulae of the Year $2016$
$j^2 = 1$, but $j \neq \pm 1$; what is $j$?
Plane intersecting line segment
The value of $\mathop{\sum\sum}_{0\leq i< j\leq n}(-1)^{i-j+1}\binom{n}{i}\binom{n}{j}$
Questions on limit superiors
The center of $A_n$ is trivial for $n \geq 4$
Is there a rule of integration that corresponds to the quotient rule?
Why $P(A) \cup P(B)$ is not equivalent to $P(A \cup B)$
Prove positive definiteness
Prime as sum of three numbers whose product is a cube
Riemann rearrangement theorem

I have faced this series several times. Could it be possible to come up with a formula for the following sequence?

$S_n= \sum_{k=1}^n k^k = 1^1 + 2^2 + 3^3 + \ldots + n^n$

If yes, then what is that and how it could be proved?

- Splitting an infinite unordered sum (both directions)
- Find the residue at $z=-2$ for $g(z) = \frac{\psi(-z)}{(z+1)(z+2)^3}$
- Prove that $\exists \{c_n\}$ monotonically increasing to $\infty$ such that $\sum_{i=1}^\infty a_nc_n$ coverges.
- Closed form for $\sum_{n=1}^{\infty}\frac{1}{\sinh^2\!\pi n}$ conjectured
- My formula for sum of consecutive squares series?
- Generalisation of alternating functions

Note: I had come through this question which I have found almost near to this series but still could not come up.

- Polygamma function series: $\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2$
- Are these two sequences the same?
- Why does $\sum_{n=1}^{\infty}\frac{\cos\frac{1}{n}}{n}$ diverge but $\sum_{n=1}^{\infty}\frac{\sin\frac{1}{n}}{n}$ converges?
- In which ordered fields does absolute convergence imply convergence?
- About the Legendre differential equation
- Very accurate approximations for $\sum\limits_{n=0}^\infty \frac{n}{a^n-1}$ and $\sum\limits_{n=0}^\infty \frac{n^{2m+1}}{e^n-1}$
- Hard Telescoping Series
- Evaluating the series $\sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} $
- Evaluating $\sum\limits_{x=1}^\infty x^2\cdot\left(\frac{1}{2}\right)^{x+1}$?
- Prove that if $\sum{a_n}$ converges absolutely, then $\sum{a_n^2}$ converges absolutely

- How can we show that $\operatorname{ord}_{p}\left(\binom{2n}n\right) \le \frac{\log 2n}{\log p}$
- If $\gcd(a,n) = 1$ and $\gcd(b,n) = 1$, then $\gcd(ab,n) = 1$.
- How to solve this sum limit? $\lim_{n \to \infty } \left( \frac{1}{\sqrt{n^2+1}}+\cdots+\frac{1}{\sqrt{n^2+n}} \right)$
- Semisimplicity is equivalent to each simple left module is projective?
- Homomorphisms and exact sequences
- Bound for the degree
- Another combinatorics problem: $\sum\limits_{k = 0}^n (-1)^k \binom{2n-k}k2^{2n-2k}=2n+1$
- $f$ be a nonconstant holomorphic in unit disk such that $f(0)=1$. Then it is necessary that
- Find where $r(t)=<t,t,t^2>$ hits the $x-y$ plane
- Finding a formula for $1+\sum_{j=1}^n(j!)\cdot j$ using induction
- Visualizing Commutator of Two Vector Fields
- Does a power series vanish on the circle of convergence imply that the power series equals to zero?
- Isomorphism types of semidirect products $\mathbb Z/n\mathbb Z\rtimes\mathbb Z/2\mathbb Z$
- The smallest symmetric group $S_m$ into which a given dihedral group $D_{2n}$ embeds
- Functional equation $f(x+y)-f(x)-f(y)=\alpha(f(xy)-f(x)f(y))$ is solvable without regularity conditions