For every positive convergent series $\sum_{n=1}^{\infty}a_{n}$, can we find another positive convergent series $\sum_{n=1}^{\infty}b_{n}$ such that $\lim_{n\rightarrow\infty}\frac{b_{n}}{a_{n}}=+\infty$? $\\$ I think the answer to the question may be no. Here is an example $\\a_{n}= \begin{cases} \frac{1}{n^2} &\text{if } n \neq k^{2}(k \in \mathbb{N^+})\\ \frac{1}{n} &\text{if } n= k^2 (k \in \mathbb{N^+}) \end{cases}$. Obviously,$\sum_{n=1}^{\infty}a_{n} \leqslant2\sum_{n=1}^{\infty}\frac{1}{n^2}$. The positive series$\sum_{n=1}^{\infty}a_{n}$ converges. I can't find the positive convergent series $\sum_{n=1}^{\infty}b_{n}$ satisfies $\lim_{n\rightarrow\infty}\frac{b_{n}}{a_{n}}=+\infty$. If you can find the positive series $\sum_{n=1}^{\infty}b_{n}$ , please tell me. Furthermore, if you can prove it is right or false, I think it will be much better.
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2Take $c_n=\sum_{k=n}^\infty a_k$ and $b_n=\sqrt{c_n}-\sqrt{c_{n+1}}$. – metamorphy Apr 13 '21 at 07:09
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1Alternatively, take $$b_n := \frac{{a_n }}{{\sqrt {\sum\limits_{k = n}^\infty {a_k } } }}.$$ – Gary Apr 13 '21 at 07:21