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this is just somthing i thought about (i dont know if there is an answer)

Let $ \sum_{n=1}^{\infty}a_n \to L_1$

($a_n $ is a positive sequence)

find a sequence $b_n$ such that:

$\lim_{n\to\infty} \dfrac{b_n}{a_n} = \infty$

and $\sum_{n=0}^{\infty}b_n = L_2$ (the series converges)

if there is no solution can this be proven?

if we add the fact that $a_n$ is monotone decending would that help?

thanks alot!

Community
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guynaa
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1 Answers1

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NOTICE

This is an answer to the original question. While I was writing the answer the OP changed the condition $a_n/b_n\to\infty$ to $b_n/a_n\to\infty$.

Take $b_n=a_n^2$. Since $\sum a_n$ converges, $a_n\to0$, and $a_n/b_n=1/a_n\to\infty$. Also, $a_n$ is bounded. Let $A$ be an upper bound. Then $0\le b_b\le A\,a_n$, so that $\sum b_n$ converges.

Julián Aguirre
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  • Notice that the problem has changed. The condition $b_n/a_n\rightarrow\infty$ is the one that shows the interesting property that for every convergent series, there is one that converges slower. – OR. Dec 20 '13 at 16:50