2
$\begingroup$

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.

$\endgroup$
4
$\begingroup$

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

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.