Prove the series $\sum_{n=1}^\infty (n(f(\frac{1}{n}) - f(-\frac{1}{n})) - 2f'(0)) $ converges where $f$ is defined on $[-1,1]$ and $f''(x)$ is continuous.
I already have a solution for this but I am not quite following it.
It uses the Taylor expansion about $0$ to get: [Note: this is the start of their solution not mine]
$f(x) = f(x) +f'(0)\frac{x}{1} + f''(0)\frac{x^2}{2} + f'''(t)\frac{x^3}{6}$
$f(-x) = f(x) - f'(0)\frac{x}{1} + f''(0)\frac{x^2}{2} - f'''(s)\frac{x^3}{6}$
For some $s,t \in [-1,1]$.
Here is my issue: I am not clear on the $s,t$ used in the third derivative. We know the second derivative is continuous, maybe the statement should be that the third derivative is continuous?
