2

I want to know the limit of the sequence that:

$$a_{1}=a>0,$$ $$a_{n+1}=\ln(1+a_{n})$$

then

$$\lim_{n\to\infty}na_{n}$$

I knew that $$\lim_{n\to\infty}a_{n}=0$$, but I can't get the value of $$\lim_{n\to\infty}na_{n}$$

This is my first question, and Sorry for my poor English.

Mayoi
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  • This has been posted here before, but I can't search right now to link it. The limit is $\frac12$, try to use Stolz-Cesaro to prove it. – Zacky Oct 10 '19 at 00:07
  • Thanks, I found the post, and The answer was 2. That post used Stolz to prove that, but I can hardly understand that prove... Is there another method to prove that the answer is 2? – Mayoi Oct 10 '19 at 00:20
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    It duplicates MSE question 1072256 "$a_{n+1}=\log(1+a_n), a_1>0.$ Then find $\lim_{n\to\infty}n\cdot a_n$". – Somos Oct 10 '19 at 00:44
  • I found it. thanks. Could anyone teach me how to get the answer without using Stolz-Cesaro theorem? – Mayoi Oct 10 '19 at 00:59
  • @Mayoi since the post is a duplicate, but you don't understand the answer from there I will suggest to edit the question so that it makes clear that you are looking for clarifications (include in the question what you understand, or don't) or alternative approaches. Afterwards your post will get reopened, but it's not a good thing to ask for alternative answers without linking to the ones that you saw already. Lastly please include some context of what you're studying, I think there's a way to solve this without Stolz-Cesaro, but it might need to use asympthotics or power series. – Zacky Oct 10 '19 at 10:27
  • thank you for your opinion. It was my first question and editing post was quite a hard work. I'll study Stolz theorem and try to understand the answer posted before. – Mayoi Oct 10 '19 at 10:59

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