Convergence proof

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March 1st 2012, 03:07 PM
Dinghy

Convergence proof

The proposition is as follows:

If $a_n\geq0$ for all $n$ $\in \mathbb N$ and there is a $p>1$ such that $$\lim_{n\to\infty} n^pa_{n}$ exists and is finite, then $\sum_{n=1}^\infty a_{n}$ converges.

I'm honestly not even sure what angle to come at this from, since I can't see what good knowing that limit exists does in trying to prove the convergence of some series. If someone could just give me a nudge in the right direction here I should be OK.
March 1st 2012, 03:37 PM
Plato

Re: Convergence proof

Quote:

Originally Posted by Dinghy

The proposition is as follows:
If $a_n\geq0$ for all $n$ $\in \mathbb N$ and there is a $p>1$ such that $$\lim_{n\to\infty} n^pa_{n}$ exists and is finite, then $\sum_{n=1}^\infty a_{n}$ converges.

This is a straightforward application of the limit comparison test.

Note that if $p>1$ then $\sum\limits_n {\frac{1}{{n^p }}}$ converges.
March 1st 2012, 03:50 PM
Dinghy

Re: Convergence proof

Quote:

Originally Posted by Plato

This is a straightforward application of the limit comparison test.

Note that if $p>0$ then $\sum\limits_n {\frac{1}{{n^p }}}$ converges.

Legitimately embarrassed about not having seen that. This problem had been talked up so much that it didn't even occur to me that it could be such a straightforward application of one of the simpler convergence tests. One thing though, when you say $p>0$ , you mean $p>1$ , right?

Thanks a lot.
March 1st 2012, 04:11 PM
Plato

Re: Convergence proof

Quote:

Originally Posted by Dinghy

you mean $p>1$ , right?

Yes, of course. Corrected!
March 1st 2012, 04:26 PM
Dinghy

Re: Convergence proof

Actually I just noticed, I don't think you can apply the limit comparison test, as technically $\sum_{n=1}^\infty a_{n}$ isn't necessarily a positive series. The proposition says that $a_{n}\geq 0$ , whereas a positive series requires that $a_{n}>0$ , so I think I'm back at square one.

edit: but I guess the Limit Comparison test doesn't strictly require it to be a positive series, just that $a_{n}\geq 0$ ...

Can't shake the feeling that my math professor is messing with me with the " $\sum_{n=1}^\infty a_{n}$ isn't necessarily a positive series!!" warning.
March 1st 2012, 04:44 PM
Plato

Re: Convergence proof

Quote:

Originally Posted by Dinghy

Actually I just noticed, I don't think you can apply the limit comparison test, as technically $\sum_{n=1}^\infty a_{n}$ isn't necessarily a positive series. The proposition says that $a_{n}\geq 0$ , whereas a positive series requires that $a_{n}>0$ , so I think I'm back at square one.k

edit: but I guess the Limit Comparison test doesn't strictly require it to be a positive series, just that $a_{n}\geq 0$ ...

Can't shake the feeling that my math professor is messing with me with the " $\sum_{n=1}^\infty a_{n}$ isn't necessarily a positive series!!" warning.

I don't understand what you are concerned about.
Look at the link I provided.
It requires that $a_n\ge 0~\&~b_n\ge 0~.$
March 1st 2012, 04:48 PM
Dinghy

Re: Convergence proof

Yeah, sorry, my Calc 2 professor defined the LCT really lazily (said that it only applies to positive series), and once I get something in my head it's really hard to get it out.

Thanks for your help.