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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.


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

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  • $\begingroup$ @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