The Maclaurin series for is (1+x)= _ n=1 ^ (-1)^ n+1 x^ n n (-1<x 1)

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

The Maclaurin series for ln⁡(1+x)\ln(1+x) is ln⁡(1+x)=∑n=1∞(−1)n+1xnn(−1<x≤1).\ln(1+x)=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}x^{n}}{n}\quad (-1<x\le 1). Define f(x)=11+xf(x)=\dfrac{1}{1+x}. (a) Derive a power series representation for f(x)f(x) centered at x=0x=0, clearly indicating the general term and the interval of convergence. (b) Use your series to evaluate the infinite series ∑n=1∞(−1)n+1n2n.\sum_{n=1}^{\infty} \frac{(-1)^{n+1}n}{2^{n}}.

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