Prove the series $ sum_ {n = 1} ^ infty (n (f ( frac {1} {n} – f ( frac {1} {n}) – 2f & # 39; (0)) $ where converges $ f $ is defined on $ (-1.1) $ and $ f & # 39; & gt; (x) $ is continuous.

I already have a solution, but I do not quite follow it.

It uses the Taylor extension over $ 0 $ to get:

$ f (x) = f (x) + f ((0) frac {x} {1} + f & # 39; # (0) frac {x ^ 2} {2} + f & # 39; & # 39; & # 39; (t) frac {x ^ 3} {6} $

$ f (-x) = f (x) – f ((0) frac {x} {1} + f & # 39; # (0) frac {x ^ 2} {2} – f & # 39; & # 39; & # 39; (s) frac {x ^ 3} {6} $

For some $ s, t in (-1,1) $,

Here is my problem: I'm not clear on that $ s, t $ used in the third derivative. We know that the second derivative is continuous, perhaps the statement should be that the third derivative is continuous?