# Apery and Fresnel join forces

Calculus Level 5

Let $$\displaystyle S(u) = \int_0^u \sin\left(\frac{\pi}{2} x^2\right) \, dx$$ be the Fresnel sine integral. If

$\sum_{n=1}^{\infty} \frac{S^2(\sqrt{2n})}{n^3}$

can be expressed in the form $$\dfrac{a}{b}\pi^c$$, where $$a$$ and $$b$$ are coprime positive integers and $$c$$ is an integer, find $$a+b+c$$.

Hint: Consider an appropriate function $$f(x) = |x|^{\sigma}$$ where $$\sigma \in \mathbb{R}$$ and apply Parseval's theorem.

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