GATE 2020 BT – Question 61

Engineering Mathematics · Calculus: Sequences and series, tests for convergence · 2 marks · Numerical answer

Find $S=1+2/3+3/9+4/27+5/81+\cdots$ to infinity, rounded to 2 decimals.

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Correct answer: 2.20 to 2.30

Explanation

The series is $\sum_{n=1}^{\infty}nr^{n-1}$ with r = 1/3. Differentiate the geometric series $\sum_{n=0}^{\infty}r^n=1/(1-r)$ for |r| < 1 to obtain $\sum nr^{n-1}=1/(1-r)^2$. Hence $S=1/(1-1/3)^2=9/4=2.25$.

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