## Archive for May 8th, 2012

### A look at a few Tripos questions V

May 8, 2012

Here is the final analysis question from 2003.

12C. State carefully the formula for integration by parts for functions of a real variable.

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

$\displaystyle f(t)=f(0)+f'(0)t+\frac 1{2!}f''(0)t^2+\dots$
$\displaystyle \dots+\frac 1{(n-1)!}f^{(n-1)}(0)t^{n-1}+\frac 1{(n-1)!}\int_0^tf^{(n)}(x)(t-x)^{n-1}dx$.

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

$\displaystyle \sum_{n=1}^\infty\frac 1{n2^n}$

converges to $\log 2$.
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