Problem with this challenging summation

Question

I'm having some trouble finding the summation of this series.

I tried all I could, but in the end the denominator is creating problem. $$ \sum_{r=0}^{n} (-1)^r \binom{n}{r}\frac{1+r\ln 10}{(1+\ln 10^n)^r} $$

The answer is zero but in my book no solution is given.

Any help would be appreciated.

Answer 1

Hint:

$$\sum_{r=0}^n \binom{n}{r} x^r = (1+x)^n$$

$$\sum_{r=0}^n r \binom{n}{r} x^r = x \frac{d}{dx} \sum_{r=0}^n \binom{n}{r} x^r = n x (1+x)^{n-1}$$