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CONTEXT: Question made up by uni lecturer

Say $\sum{a_n}$ is convergent. Does this then mean that $\lim_{n\to \infty}|\frac{a_{n+1}}{a_n}|\neq1$?

I know that the ratio test can be used to prove that a series converges, but I feel that there surely exists a convergent series where $\lim_{n\to \infty}|\frac{a_{n+1}}{a_n}|=1$ as it may have already passed another test proving its convergence.

Can anyone think of such a series?

TIA

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Consider $\sum \dfrac1{n}$ and $\sum\dfrac 1{n^2}$. What does the ratio test give in each case?

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  • $\begingroup$ The harmonic series is divergent yet the ratio test gives an answer of 1 (so this proves the contrapositive of my question). While the second series converges and the ratio test gives it an answer of 1 (this proves my question). Thanks! $\endgroup$ – Ruby Pa May 5 '19 at 4:09
  • $\begingroup$ No. An example isn't a proof. You can have limit 1 with both convergent and divergent series. $\endgroup$ – Ted Shifrin May 5 '19 at 4:10
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    $\begingroup$ Oops, I meant that it proves my suspicions. The second series converges and the ratio test gives it an answer of 1, which proves the negation of the statement. $\endgroup$ – Ruby Pa May 5 '19 at 4:16

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