State carefully the formula for integration by parts for functions of a real variable.

Let $f:(-1,1) \rightarrow \mathbb{R}$ be infinitely differentiable. Prove that for all $n \geqslant 1$ and all $t \in(-1,1)$,

$f(t)=f(0)+f^{\prime}(0) t+\frac{1}{2 !} f^{\prime \prime}(0) t^{2}+\ldots+\frac{1}{(n-1) !} f^{(n-1)}(0) t^{n-1}+\frac{1}{(n-1) !} \int_{0}^{t} f^{(n)}(x)(t-x)^{n-1} d x .$

By considering the function $f(x)=\log (1-x)$ at $x=1 / 2$, or otherwise, prove that the series

$$\sum_{n=1}^{\infty} \frac{1}{n 2^{n}}$$

converges to $\log 2$.