Quick question about Ratio Test for Series Convergence

[ColtonCM]

Homework Statement

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This is the question I have (from a worksheet that is a practice for a quiz). Its a conceptual question (I guess). I understand how to solve ratio test problems.

"Is this test only sufficient, or is it an exact criterion for convergence?"

Homework Equations

Recall the ratio-test: If {a_n}_n∈N is a positive sequence and there is a number a < 1 such that eventually a_n+1 ≤ a then the series is convergent. If, eventually, a_n+1 ≥ 1 then the series is divergent.

The Attempt at a Solution

I would assume that it would be considered "only sufficient," since if the result yields a ratio of one, convergence cannot be determined, thus it is not an absolute criterion.

Would this line of reasoning be correct?

Thanks,

Colton

 

[fourier jr]

that sounds right. it's sufficient but not necessary because there are other ways to determine whether or not a series converges eg $\sum_{n} \frac{1}{n^{2}}$ is known to converge but the ratio test doesn't give any information about it.

 

[ColtonCM]

Sounds good, thanks!