Analysis | Part IA, 2003

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

Let f:(1,1)Rf:(-1,1) \rightarrow \mathbb{R} be infinitely differentiable. Prove that for all n1n \geqslant 1 and all t(1,1)t \in(-1,1),

f(t)=f(0)+f(0)t+12!f(0)t2++1(n1)!f(n1)(0)tn1+1(n1)!0tf(n)(x)(tx)n1dx.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(1x)f(x)=\log (1-x) at x=1/2x=1 / 2, or otherwise, prove that the series

n=11n2n\sum_{n=1}^{\infty} \frac{1}{n 2^{n}}

converges to log2\log 2.

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