Question

Show that if there is an absolutely convergent series Sum(k=1,Infinity,ak) then there exists an absolutely convergent series Sum(k=1,Infinity,bk) such that as lim(k->Infinity) ak/bk -> 0

Answer 1

(Restated)

Show that if there is an absolutely convergent series
\[\sum_{k=1}^{\infty} a _{k}\]
then there exists an absolutely convergent series
\[\sum_{k=1}^{\infty} b _{k}\]
such that \[\lim_{k \rightarrow \infty} \rightarrow 0\]

Answer 2

\[\lim_{k \rightarrow \infty} \frac{ a _{k} }{ b _{k} } \rightarrow 0\]

Answer 3

Would \[b _{k}=\sum_{j=1}^{k}a _{j}\] work?

I'm thinking this would allow the limit to approach 0, but I'm not sure if bk would be absolutely convergent or not...

Answer 4

Are you there Zarkon?