I make mistake so I get different

$\sum_{n=1}^{\infty}\left(\sum_{k=0}^{n}\dfrac{(-1)^k}{k+1} -\sum_{m=0}^{\infty}\sum_{r=0}^{m}\dfrac{(-1)^r(-1)^m}{r+1}\binom{m}{r}\right)$

The closed form of the sum above can be expressed as $\ln a -\dfrac{b}{c}$ , where $a$ , $b$ , and $c$ are positive integers with $b$ and $c$ being primes. Find the value of $a+b+c$ .