The GATE Grind

GATE 2026 EC – Question 36

Engineering Mathematics · Complex Analysis · 2 marks · Multiple choice

Consider the two series, $S_A$ and $S_B$, where $S_A=\sum_{n=1}^{\infty}\frac{n^2}{2^n}$ and $S_B=1+\frac12+\frac18+\frac1{16}+\frac1{64}+\frac1{128}+\frac1{512}+\cdots$. Which of the following statements is correct for the two given series?

  1. Both $S_A$ and $S_B$ converge.
  2. Neither $S_A$ nor $S_B$ converges.
  3. $S_A$ converges but $S_B$ does not converge.
  4. $S_B$ converges but $S_A$ does not converge.

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Show answer and explanation

Correct answer: (A) Both $S_A$ and $S_B$ converge.

Explanation

By the ratio test, $S_A$ has ratio $\to1/2<1$, so it converges (to 6). The terms of $S_B$ are $2^{-0},2^{-1},2^{-3},2^{-4},2^{-6},\dots$, each bounded by a geometric series with ratio 1/2 (grouped in pairs), so $S_B$ also converges.