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Understanding when $$\lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} \frac{a_0 + a_1+ \cdots + a_n}{b_0+b_1+ \cdots + b_n}.$$

That is, supposing the first limit exists, what properties must be satisfied by the sequences $\{a_n\}_{n \ge 0}, \{b_n\}_{n \ge 0}$ such that we may conclude that the above equality holds? I tried using the inequality $\frac{a_n}{b_n} + \frac{a_{n-1}}{b_{n-1}} < \frac{a_n+a_{n-1}}{b_n + b_{n-1}}$, which holds when our sequences consist of positive real numbers, having in mind the notion of applying induction but didn't get anywhere with that idea. This isn't a homework assignment, I just want to know the answer.

Lorenzo B.
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Sid
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  • Oh cool I'll check out the Stolz-Cesaro Theorem. – Sid Jul 29 '17 at 23:21
  • Basically, let $sa_n=a_0+\dots+a_n$ and $sb_n=b_0+\dots+b_n$, then apply Stolz-Cesaro. – Simply Beautiful Art Jul 29 '17 at 23:21
  • @SimplyBeautifulArt Thanks for the help. Follow up question: Why, assuming the first limit exists, does $\lim_{n \to \infty} \frac{a_0+a_1+ \cdots + a_n}{b_0+b_1+\cdots+b_n} = \lim_{x \to 1}\frac{a_0+a_1x+a_2x^2+\cdots}{b_0+b_1x+b_2x^2+\cdots}$? It's intuitively "obvious" but I can't formulate it in terms of epsilons and deltas. – Sid Jul 30 '17 at 00:06
  • See Section 2 of this paper for my motivations https://www.google.com/search?q=ramanujan+on+certain+arithmetical+functions&oq=ramanujan+on+certain+arithmetical+functions&gs_l=psy-ab.3...21394.21628.0.21762.3.3.0.0.0.0.0.0..0.0....0...1.1.64.psy-ab..3.0.0.-jnVobcfiXM – Sid Jul 30 '17 at 00:58
  • Hm, I'm not entirely sure how to prove it. I get the feeling it follows from Abel's theorem or something, assuming that $\sum a_k$ and $\sum b_k$ exist. – Simply Beautiful Art Jul 30 '17 at 12:28

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