A calculus problem by 敬全 钟

Calculus Level 4

11+121314+15+161718+\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 π+ln(a)a\dfrac{\pi+\ln (a)}{a}, where aa is a positive integer, find aa.

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