sequences and series - What is the sum of the first 21 numbers of the formula below?

I know how to solve the geometric progression and arithmetic progression but this one seems strange to me, it isn't even a harmonic serie, Any help for solving it would be appreciated.

$$\sum_{n=1}^{21}\frac{3n+1}{n^{n+1}}$$

I have simplified the formula to this:

$\frac{3n}{n^{n+1}}$ --> $\frac{3}{n^{n}}$ --> $3\frac{1}{n^{n}}$

and $\frac{1}{n^{n+1}}$
so it would be:

$3\frac{1}{n^{n}}$ + $\frac{1}{n^{n+1}}$

I there would be a formula for calculating the sum of $\frac{1}{n^{n}}$ the problem would have been solved.