Convergence of the ratio of two infinite series.

by isentr0pic   Last Updated May 28, 2018 10:20 AM

I'm currently working through problems given in Mathematical Methods for Physicists by Arken et al. I'm having a bit of trouble resolving a problem that comes up in Chapter 1. It is as follows:

If $\lim_{n\rightarrow\infty} \frac{b_n}{a_n} = K$, a constant with $0 < K < \infty$, show that $\sum_{n}b_n$ converges or diverges with $\sum a_n$.

Provided Hint: If $\sum a_n$ converges, rescale $b_n$ to $b_n' = \frac{b_n}{2K}$. If $\sum a_n$ diverges, rescale to $b_n'' = \frac{2b_n}{K}$.

Plugging everything in, I'm left with the following expression: $$ \lim_{n\rightarrow\infty} \frac{b_n}{a_n} = K \longrightarrow \lim_{n\rightarrow\infty} \frac{b_n'}{a_n} = \frac{1}{2} \\ $$

From here, I would usually proceed to split the limit and rearrange like so: $$ \frac{\lim_{n\rightarrow\infty}b_n'}{\lim_{n\rightarrow\infty}a_n} = \frac{1}{2} \\ \lim_{n\rightarrow\infty}b_n' = \frac{1}{2} \lim_{n\rightarrow\infty} a_n $$

In this case, I realise I can't do that because of the fact that under the assumption that $\sum a_n $ converges, $\lim_{n\rightarrow\infty} a_n$ must be equal to zero by the limit test.

Where have I messed up on my reasoning?


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