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alexmahone
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Is the converse of the ratio test true?
Krizalid said:I don't think so. I think you can construct an easy counterexample. Care to imagine one?
The "ratio test" says that if $lim \frac{a_{n+1}}{a_n}< 1$ then $\sum a_n$ converges.Alexmahone said:0+0+0+... converges but the ratio is not defined.
I wonder if there are any non-trivial counterexamples.
Maybe...Alexmahone said:0+0+0+... converges but the ratio is not defined.
I wonder if there are any non-trivial counterexamples.
HallsofIvy said:Find a convergent series such that that limit is 1.
The converse of the ratio test is a theorem that states if the limit of the absolute value of the ratio of successive terms in a sequence is less than 1, then the series converges.
The ratio test is used to determine the convergence or divergence of a series, while the converse of the ratio test is used to prove the convergence of a series.
The converse of the ratio test is important in proving the convergence of series that do not fit the criteria of the ratio test, but still converge.
No, the converse of the ratio test can only be used to prove the convergence of a series, not the divergence. It is possible for a series to satisfy the criteria of the converse of the ratio test and still diverge.
The converse of the ratio test is used in various fields of science and engineering, such as physics, chemistry, and computer science, to prove the convergence of series and ensure the accuracy of calculations and simulations.