I've encountered a recommended practice proof that I'd like some assistance in starting.

Suppose that $\sum_{i=1}^\infty an$ and $\sum_{i=1}^\infty bn$ are both series with all positive terms and $\sum_{i=1}^\infty bn$ is convergent. Prove that if $lim_{n\to \infty} an/bn$ = 0, then $an$ converges.

This appears to be the limit comparison test for series, but the requirement for 0 < c < inf is not met. Any ideas would be appreciated.


Hint: $$\lim_{n \to \infty} \dfrac{a_n}{b_n} = 0 \qquad \Longrightarrow \qquad \dfrac{a_n}{b_n} < 1$$ for large $n$.

  • $\begingroup$ From this I think i could say bn > an, such that by the comparison test, an must also converge. Am i on the right track or missing the point? $\endgroup$ – user105781 Nov 6 '13 at 22:40
  • $\begingroup$ could anyone confirm or deny this? $\endgroup$ – user105781 Nov 7 '13 at 1:33
  • $\begingroup$ You are absolutely right. $\endgroup$ – njguliyev Nov 7 '13 at 19:14

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