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Let $\{a_n\}_{n\geq 1}$ be a sequence of strictly decreasing positive numbers i.e., $$a_1>a_2>a_3>\cdots$$

Then which of the following is always true?

  • $\lim _{n\rightarrow \infty} a_n=0$
  • $\lim _{n\rightarrow \infty} \frac{a_n}{n}=0$
  • $\sum_{n=1}^{\infty}\frac{a_n}{n}$ converges.
  • $\sum_{n=1}^{\infty}\frac{a_n}{n^2}$ converges.

I guess all options are correct but i could not give a concrete proof.

For third bullet, Take $b_n=\frac{a_n}{n}$ and look for ratio test :

$\lim_{n\rightarrow \infty}\frac{b_{n+1}}{b_n}=\lim_{n\rightarrow \infty}\frac{a_{n+1}}{n+1}.\frac{n}{a_n}\rightarrow 0$ Thus $\sum_{n=1}^{\infty}\frac{a_n}{n}$ converges.

For fourth bullet, I would just use :

As $a_n$ are positive and $\sum_{n=1}^{\infty}\frac{a_n}{n}$ converges with $\frac{a_n}{n^2}<\frac{a_n}{n}$.

we would say $\sum_{n=1}^{\infty}\frac{a_n}{n^2}$ converges.

I can not convince myself that this is a proof.

So, please help me to make this a bit more clear.

Thank you :)

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  • 4
    $\begingroup$ Hint: How does $a_n=42+\frac{1}{n}$ behave? $\endgroup$ – Peter Košinár Dec 25 '13 at 11:10
  • $\begingroup$ Ah... May be that is the reason i could not come up with result $\lim a_n=0$ :D Thank you so much.... $\endgroup$ – user87543 Dec 25 '13 at 11:12
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The sequence $a_n=1+\frac{1}{n}$ shows that the first and the third point are not true.

the inequalities $$0\leq \frac{ a_n}{ n}\leq \frac {a_1}{ n}$$ prove that the second point is true and the inequalities $$0\leq \frac{ a_n}{ n^2}\leq \frac {a_1}{ n^2}$$ prove that the last point is also true.

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  • $\begingroup$ You're welcome. $\endgroup$ – user63181 Dec 25 '13 at 12:29
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The correct options are the second and the fourth (Peter’s comment contains a counterexample to both the first and third options). You can deduce the second option from the fourth.

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  • $\begingroup$ @SamiBenRomdhane corrected, thanks. $\endgroup$ – Ewan Delanoy Dec 25 '13 at 11:28
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For the third bullet point you can also consider this

$a_n=\frac{n}{2n-1}$

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  • $\begingroup$ why would you think this is worth considering? Is there something more interesting than other examples? I would like to know more about this... $\endgroup$ – user87543 Dec 25 '13 at 15:24
  • $\begingroup$ this will disprove bullet point 3 $\endgroup$ – tattwamasi amrutam Dec 25 '13 at 15:43
  • $\begingroup$ Yes Yes i understand that this would disprove third bullet point but i was expecting you want to say that this is more simple/interesting than other examples... Thank you any ways :) $\endgroup$ – user87543 Dec 25 '13 at 15:50
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    $\begingroup$ interesting in the sense that it disproves 1and 3 simultaneosuly $\endgroup$ – tattwamasi amrutam Dec 26 '13 at 3:37
  • $\begingroup$ valid point :) Thank you :) $\endgroup$ – user87543 Dec 26 '13 at 4:38

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