Is there a way to solve the following series?

Question

I was solving a question related to permutation and combination and I came across the following series. $$\binom{n}{0}k^0+2\binom{n}{1}k^1+3\binom{n}{2}k^2+...+(n+1)\binom{n}{n}k^n$$

Is there a way to solve the series?

Answer 1

Hint it is the derivative of $k(1+k)^n$

Answer 2

For $r\ge1,$

$$(r+1)\binom nr=r\binom nr+\binom nr$$

$$=\binom nr+r\cdot\dfrac{n\cdot(n-1)!}{r\cdot(r-1)!\{n-1-(r-1)\}!}=\binom nr+n\binom{n-1}{r-1}$$

$$\implies\sum_{r=0}^n(r+1)\binom nrk^r=\sum_{r=0}^n\binom nrk^r+nk\sum_{r=1}^n\binom{n-1}{r-1}k^{r-1}$$

Now use $\displaystyle(a+b)^m=\sum_{r=0}^m\binom mra^{m-r}b^r$