# Thread: How to prove that a limit exists?

1. ## How to prove that a limit exists?

Hello.

I have to prove that an actual limit exists while n is approaching infinity.
I do NOT need to get to the exact limit itself.

1. It is easy to understand (and prove), that sequence never decreases. That's already half of the job.
2. I need to make some manipulations that would lead to any specific number being greater than my sequence (irrespective of natural n value).

I get no further than this, a remaining product is a real headache.
Thank you for ideas and sorry for possibly broken English

2. Originally Posted by Pranas
Hello.

I have to prove that an actual limit exists while n is approaching infinity.
I do NOT need to get to the exact limit itself.

1. It is easy to understand (and prove), that sequence never decreases. That's already half of the job.
2. I need to make some manipulations that would lead to any specific number being greater than my sequence (irrespective of natural n value).

I get no further than this, a remaining product is a real headache.
Thank you for ideas and sorry for possibly broken English
$\lim\limits_{n\to\infty}\prod\limits^n_{k=1}\frac{ 2^k+1}{2k}=\prod\limits^\infty_{k=1}\left(1+\frac{ 1}{2^k}\right)$ . Applying logarithm you get that the infinite product converges

iff the infinite series $\sum\limits^\infty_{k=1}\log\left(1+\frac{1}{2^k}\ right)$ converges (the logarithm's base never minds, of course).

Lemma: if $\{a_n\}$ is a positive sequence, the series $\sum\limits^\infty_{k=1}\log\left(1+a_k\right)$ converges iff the series $\sum\limits^\infty_{k=1}a_k$ converges.

Proof sketch: assuming $a_k\xrightarrow [k\to\infty]{}0$ , we get what we want from the limit test for positive series, since $\frac{\log(1+a_k)}{a_k}\xrightarrow [k\to\infty]{}1\,\,\,\,\square$

Tonio

3. Thank you very much, I was not aware of the lemma.