icantadd
- 109
- 0
Homework Statement
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
Homework 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.
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