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Suppose $\sum_{n=1}^\infty a_n = A$ exists and $\left(b_n\right)$ is a monotone sequence with limit $B$. $$\text{Prove }\sum_{n=1}^\infty a_n b_n \text{ converges?}$$

Can this be done using Dirichlet's test? I thought the condition was that the monotone sequence had to have $\lim_{n\to\infty}b_n = 0$.

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$\sum a_n b_n = \sum a_n (b_n - B) + \sum a_nB =$ convergent$+ AB < \infty$

Just a guess.

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    $\begingroup$ Thanks! That makes sense!! I thought maybe something like that but I wasn't sure. $\endgroup$ – math_rookie Mar 1 '15 at 22:55
  • $\begingroup$ @math_rookie I'm just guessing. This is the first time in my life that I have used Dirichlet's test. Wait for another answer. Maybe + B instead? $\endgroup$ – BCLC Mar 1 '15 at 22:56
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    $\begingroup$ You could argue that $\lim_{n\to\infty}(b_n - B)=0$ and $b_n-B$ is also monotone, therefore the sequence $c_n=b_n-B$ satisfies the conditions of Dirichlet's test $\endgroup$ – math_rookie Mar 1 '15 at 23:02
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If $b_n$ is monotone increasing and bounded above, let $B=\sup_n b_n$. Then for any $N$,

$$\sum_{n=1}^N a_nb_n\leqslant B\sum_{n=1}^N a_n\to BA $$ as $n\to\infty$. (If $b_n$ is monotone decreasing and bounded below, just use $\inf_n b_n$.)

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    $\begingroup$ Are you assuming that $a_n b_n\ge0$ for all n? $\endgroup$ – user84413 Mar 2 '15 at 0:23

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