GATE EC 2022 Set 1 — Question 45
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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
(B) c = 2, d = 1; (D) c = 1, d = -2
Solution
The given series is .
We can use the Ratio Test for convergence. Let .
Then, .Taking the limit as :
.According to the Ratio Test:
Here, , so the Ratio Test is inconclusive. We substitute into the series:
.
This is the harmonic series, which is known to diverge.Option B:
Here, . Since , . The series converges.Option C:
Here, . Since , . The series diverges.Option D:
Here, , so the Ratio Test is inconclusive. We substitute into the series:
.
This is a p-series with . Since , this series converges.Therefore, the series converges for combinations in options B and D.The final answer is
We can use the Ratio Test for convergence. Let .
Then, .Taking the limit as :
.According to the Ratio Test:
- If , the series converges absolutely.
- If , the series diverges.
- If , the test is inconclusive, and other tests must be used.
Here, , so the Ratio Test is inconclusive. We substitute into the series:
.
This is the harmonic series, which is known to diverge.Option B:
Here, . Since , . The series converges.Option C:
Here, . Since , . The series diverges.Option D:
Here, , so the Ratio Test is inconclusive. We substitute into the series:
.
This is a p-series with . Since , this series converges.Therefore, the series converges for combinations in options B and D.The final answer is
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