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Assume $\{a_n\}$ and $\{b_n\}$ are two positive series such that $$\sum_{n}a_n=\sum_n b_n=1.$$ Assume also for all $n$, $\sum_{k\geq n}a_k\leq \sum_{k\geq n}b_k$ and $$\lim_{n\rightarrow +\infty}\frac{\sum_{k\geq n}a_k}{\sum_{k\geq n}b_k}=0.$$ Do we have $$\lim_{n\rightarrow+\infty}\frac{a_n}{b_n}=0?$$

It seems to me this is very much like discrete version of L'Hospital rule.

user111463
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Hint: Consider for $n\ge1$, $$ a_n=2^{-n} $$ and $$ b_n=\left\{\begin{array}{} 3^{-(n+1)/2}&\text{if $n$ is odd}\\ 2\cdot5^{-n/2}&\text{if $n$ is even} \end{array}\right. $$

The discrete analog of L'Hôpital would be Stolz-Cesáro. See this question.

What is suggested in the current question would be the analog of "if the limit of the ratio of two functions exists, then it is equal to the limit of the ratio of the derivatives of those functions".

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robjohn
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  • What happens with the fraction $\frac{\sum_{k \ge n} a_k}{\sum_{k \ge n} b_k}$ – kmitov May 29 '15 at 08:33
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    Since there was not much work shown, this was just a hint. However, $$\sum_{k\gt n}a_k=2^{-n}$$ and $$\sum_{k\gt n}b_k=\frac12\cdot3^{-\lfloor(n+1)/2\rfloor}+\frac12\cdot5^{-\lfloor n/2\rfloor}$$ – robjohn May 29 '15 at 08:48