Let be a differentiable function satisfying and

MATHEMATICS · Subjective · AoPS · Mathematics · Sequences & Series · Sequences & Series

Problem

Let

f(x)

be a differentiable function satisfying

f(x) + f\left(x + \frac{1}{2}\right) = 1, \forall x \in \mathbb{R}

and

g(x) = \int_{0}^{x} f(t) dt

. If

g(1) = 1

, then the value of

\sum_{n=2}^{\infty} \left( \frac{8}{\sum_{k=1}^{n} (g(x+k^2) - g(x+k))} \right)

is.