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 \(a\).