A calculus problem by 敬全钟

Calculus Level 4

$\large \dfrac{1}{1}+\dfrac{1}{2}-\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{5}+\dfrac{1}{6}-\dfrac{1}{7}-\dfrac{1}{8}+\cdots$

If the closed form of the sum above can be expressed as $\dfrac{\pi+\ln (a)}{a}$ , where $a$ is a positive integer, find $a$ .

A calculus problem by 敬全 钟