For every real number satisfying

MATHEMATICS · Multi-part · CBSE · Mathematics · Sequences and Series · Sequences and Series

Problem

For every real number xx satisfying ∣x∣<3|x|<3, define the series S(x)=∑n=0∞(−1)n(x3)nS(x)=\displaystyle\sum_{n=0}^{\infty} (-1)^{n}\left(\dfrac{x}{3}\right)^{n}. Let I=∫01S(x) dxI=\displaystyle\int_{0}^{1} S(x)\,dx. (a) Rewrite S(x)S(x) as a single elementary function of xx valid for ∣x∣<3|x|<3. Show all steps that justify your closed‐form expression. (b) Evaluate the integral II exactly. (c) Explain why term-by-term integration of the original series ∑n=0∞(−1)n(x3)n\sum_{n=0}^{\infty} (-1)^{n}\left(\dfrac{x}{3}\right)^{n} on the interval [0,1][0,1] is permissible.

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