# If $\sum_{n=0}^\infty a_n$ is convergent and $a_n \sim b_n$, is $\sum_{n=0}^\infty b_n$ also convergent?

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?

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