Allied & ICAR Exam Bank • ALLIED_2658 NAT • 2 Marks

Section 1: Engineering Mathematics

Calculus

Subtopic: Sequences and series – infinite series, tests for convergence

Question Statement:

The radius of convergence $R$ of the power series $\sum_{n=1}^\infty \frac{(2x)^n}{n^2}$ is:

Answer Key:

0.5

Step-by-Step Solution & Explanation

Using ratio test: $\lim_{n \to \infty} |\frac{2^{n+1} x^{n+1}}{(n+1)^2} \frac{n^2}{2^n x^n}| = 2 |x| < 1 \implies |x| < 1/2$. Thus $R = 0.5$.
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