GATE 2022 EC – Question 45

Engineering Mathematics · Calculus · 2 marks · Multiple select

Consider the following series:

$$\sum_{n=1}^{\infty}\frac{n^d}{c^n}$$

For which of the following combinations of $c,d$ values does this series converge?

$c=1,\ d=-1$
$c=2,\ d=1$
$c=0.5,\ d=-10$
$c=1,\ d=-2$

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Show answer and explanation

Correct answer: (B) $c=2,\ d=1$; (D) $c=1,\ d=-2$

Explanation

For $c=1$ the series is $\sum n^d$, which converges only if $d<-1$: A (the harmonic series) diverges and D converges. For $c>1$ the geometric factor $c^{-n}$ dominates any power of $n$, so B converges. For $c=0.5$ the term is $n^{-10}2^n\to\infty$, so C diverges.