If $\sum_{n=0}^\infty a_n$ is convergent and $a_n \sim b_n$ (e.g. $\lim_{n\to\infty} a_n/b_n = 1)$, is $\sum_{n=0}^\infty b_n$ also convergent?

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    $\begingroup$ Do you assume $a_n > 0$? $\endgroup$ – Daniel Fischer Oct 25 '16 at 16:11
  • $\begingroup$ @DanielFischer I'm not entirely sure of the consequences that assumption would make, so I'm inclined to say "maybe". $\endgroup$ – Frank Vel Oct 25 '16 at 16:14
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    $\begingroup$ If you don't assume an eventually fixed sign, consider $b_n = \dfrac{(-1)^n}{\sqrt{n+1}} + \dfrac{1}{n+1}$. $\endgroup$ – Daniel Fischer Oct 25 '16 at 16:16

Assuming that all the terms are positive, the quotient $b_n/a_n$ converges to $1$ and hence is bounded, say by $k$. Then, $$\sum b_n\le\sum ka_n=k\sum a_n$$


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