What can we say about convergence of $ \frac{\sum_{s=1}^k a_s^2b_s}{\sum_{s=1}^k a_s} $ when $\sum a_n=\infty$ and $(b_n)\geq 0$?

Asked May 16 '19 at 02:46

Active May 16 '19 at 06:31

Viewed 31 times

My original question is Assumption on Mirror Descent convergence? but I realized that it boils down into the following question:

Suppose $\sum a_n =\infty$ as $n \rightarrow \infty$ where $(a_n)$ is positive and $a_n \rightarrow 0$. Also, $(b_n)$ is non-negative and bounded.

What can we say about convergence of $ \frac{\sum_{s=1}^k a_s^2b_s}{\sum_{s=1}^k a_s} $?

edited May 16 '19 at 06:31

asked May 16 '19 at 02:46

Is $(a_n)$ a non convergent sequence or is $\sum a_n$ is a divergent series? – Kavi Rama Murthy May 16 '19 at 05:50
@Kavi Rama Murthy: $\sum{a_n}=\infty$ – May 16 '19 at 06:21
You have clearly said $(a_n)$ is a 'divergent sequence'. Please try to distinguish between sequences and series. – Kavi Rama Murthy May 16 '19 at 06:23
@Kavi Rama Murthy: My fault. I will edit the statement. – May 16 '19 at 06:26
There is another hypothesis in your link that says $a_n \to 0$. Please be careful in posting your question. Any 'little' hypothsis can make a big difference. – Kavi Rama Murthy May 16 '19 at 06:30
@Kavi Rama Murthy: You are right. I will be. – May 16 '19 at 06:33
you are missing the hypothesis that $b_n$ is bounded.. This question was completely answered in your first thread.. https://math.stackexchange.com/questions/3226420/assumption-on-mirror-descent-convergence – xel May 17 '19 at 14:26

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