GATE EC 2022 Set 1 — Question 55
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Engineering Mathematics → Calculus → Sequences, Series & Convergence
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Question
Consider the following series: For which of the following combinations of values does this series converge?
Correct answer
7.9 to 8.1
Solution
The given series is .
We use the Ratio Test for convergence. Let .
Then .
The limit as is .
For the series to converge, we need , which means , or .Let's check the options:
(A) . Here . The ratio test is inconclusive. The series becomes . This is the harmonic series, which diverges. So (A) is incorrect.
(B) . Here . The series converges by the ratio test. So (B) is correct.
(C) . Here . The series diverges by the ratio test. So (C) is incorrect.
(D) . Here . The ratio test is inconclusive. The series becomes . This is a p-series with , which converges. So (D) is correct.Thus, both (B) and (D) are correct combinations for which the series converges.
We use the Ratio Test for convergence. Let .
Then .
The limit as is .
For the series to converge, we need , which means , or .Let's check the options:
(A) . Here . The ratio test is inconclusive. The series becomes . This is the harmonic series, which diverges. So (A) is incorrect.
(B) . Here . The series converges by the ratio test. So (B) is correct.
(C) . Here . The series diverges by the ratio test. So (C) is incorrect.
(D) . Here . The ratio test is inconclusive. The series becomes . This is a p-series with , which converges. So (D) is correct.Thus, both (B) and (D) are correct combinations for which the series converges.
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