If $a_n,b_n\geq 0,$ then does $\lim_{n\to\infty} \frac{a_n}{b_n} = \gamma \implies \lim_{n\to\infty} \left(\frac{\sum a_n}{\sum b_n}\right)=\gamma\ ?$

Asked Dec 17 '21 at 22:05

Active Dec 17 '21 at 22:05

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More precisely, if $\ (a_n)\ $ and $\ (b_n)\ $ are non-negative real sequences with $\ a_n\not\to 0,\ b_n\not\to 0\ $ and $\ \displaystyle\lim_{n\to\infty} \left( \frac{a_n}{b_n}\right) = \gamma,$ then is it true that $\ \displaystyle\lim_{n\to\infty} \left(\frac{\displaystyle\sum_{i=1}^{n} a_i}{\displaystyle\sum_{i=1}^{n} b_i}\right)=\gamma\quad ?$

This feels like it should be true, but not sure if there is a straightforward way to prove it.

asked Dec 17 '21 at 22:05

Adam Rubinson

20,052

Stolz–Cesàro theorem – Martin R Dec 17 '21 at 22:10
@MartinR I am not well-versed with that theorem. Upon first glance, I don't see how it is relevant. Please can you elaborate? – Adam Rubinson Dec 17 '21 at 22:12
1

You apply that theorem to $A_n = \sum_{i=1}^{n} a_i$ and $B_n = \sum_{i=1}^{n} b_i$. – Martin R Dec 17 '21 at 22:15

If $a_n,b_n\geq 0,$ then does $\lim_{n\to\infty} \frac{a_n}{b_n} = \gamma \implies \lim_{n\to\infty} \left(\frac{\sum a_n}{\sum b_n}\right)=\gamma\ ?$

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