(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that if:

[tex]lim_{k\to\infty}b_k\to+\infty[/tex],

[tex]\sum_{k=1}^\infty a_k[/tex] converges and,

[tex]\sum_{k=1}^\infty a_k b_k[/tex] converges, then

[tex]lim_{m\to\infty} b_m \sum_{k=m}^\infty a_k = 0[/tex]

2. Relevant equations

3. The attempt at a solution

I only have an idea why this is true--[tex]\sum a_k[/tex] converges, so the tails become small, and [tex]\sum a_k b_k[/tex] converges, so [tex]a_k[/tex] shrinking dominates [tex]b_k[/tex] blowing up. I know that [tex]|b_k| < \epsilon / |a_k|[/tex] for any epsilon and largek, and I know that [tex]\left| \sum_{m=k}^\infty a_m \right| < \epsilon[/tex] for largek, but I can't find the way to combine these two statements.

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# Limit of a convergent series and a divergent sequence

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