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This question already has an answer here:

I've been trying to solve these two problems regarding limits and series, yet I couldn't find a proper solution:

  1. Let $\sum_{n=0}^\infty x_n$ be a real number series, which is convergent. Show that $\mathop{\underline{\lim}}_{n \to \infty} nx_n = 0$. Additionally, knowing that $x_{n+1} < x_n$, find that $\exists \ lim_{n\rightarrow \infty}nx_n = 0$.

  2. Let $(a_n)_{n \ge 0}$ be a series of $R$ defined as follows: $a_{n+1} = \sin a_n, \ \forall n\in N$ and $a_0 \in (0,\frac{\pi}{2})$.

a) Find $lim_{n\rightarrow \infty}a_n$ and $lim_{n\rightarrow \infty}\frac{1}{na_n^2}$.

b) Find the nature of the following series $\sum_{n=0}^\infty a_n^\alpha$, where $\alpha$>0.

Any help with these two will be much appreciated. Thank you all in advance!

PS: If someone could give step-by-step proof, that would be great.

PPS: This question has been marked as possible duplicate of Let $(x_n)\downarrow 0$ and $\sum x_n\to s$. Then $(n\cdot x_n)\to 0$. Even though the given link adresses a similar problem, it is not the same as the one I have posted. Above all, my post consists of two separate problems, whilst the one given in the link has only one.

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marked as duplicate by Masacroso, John B, Behrouz Maleki, hardmath, Daniel W. Farlow Feb 3 '17 at 22:34

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ @Masacroso It's not quite the same thing, leaving aside the fact that I'm asking 2 (two) separate questions. $\endgroup$ – theSongbird Feb 3 '17 at 8:02
  • $\begingroup$ The first one isn't correct. Consider $\sum_{n=0}^\infty \frac{(-1)^n}{n+1}$. $\endgroup$ – Tim B. Feb 3 '17 at 8:12
  • $\begingroup$ What does the underline of $\lim$ mean in the first question? $\endgroup$ – 5xum Feb 3 '17 at 8:15
  • $\begingroup$ @5xum: That's sometimes used as notation for $\liminf$. (Though that doesn't make the claim true; e.g. it fails for the alternating harmonic series). $\endgroup$ – Henning Makholm Feb 3 '17 at 8:18
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    $\begingroup$ @MattB: You should not ask two separate questions in a single post. Doing so does not play well together with the organization and structure of the site. $\endgroup$ – Henning Makholm Feb 3 '17 at 8:23