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

Assume [tex]\sum_{1}^{\infty} a_n[/tex] is absolutely convergent and {bn} is bounded. Prove [tex]\sum_{1}^{\infty} a_n * b_n[/tex] is absolutely convergent

2. Relevant equations

A series is absolutely convergent iff the sum of | an | is convergent

A series is convergent if for every e there is an N and P such that for all n >= N, for all p >0,

| S_{n+p} - S{n-1} | < e. S_k is the kth partial sum, and this basically says that the sequence of partial sum must be cauchy.

3. The attempt at a solution

because the series of an is abosultely convergent | |a_{n+p}| - |a_{n-1}| | < e

and bn is bounded means | b_n| < M

[tex] \left| \left| a_{n+p} b_{n+p} \right| - \left| a_{n-1}b_{n-1} \right| \right| \leq \left| a_{n+p} b_{n+p} \right| + \left| a_{n-1} b_{n-1} \right| \leq M \left| a_{n+p} \right| + M \left| a_{n-1} \right| < \frac{Me}{2M} + \frac{Me}{2M} = e [/tex]

done

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# Homework Help: The product of absconverg series and bounded seq is absolutely convergent

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