convergent sequances

lomos_lupin

What are the suffice and necessary conditions for a sequance like $\{a_i\}$ such that

1. $\sum_{i=1}^{\infty} x_i=$ convergent.

2. $\sum_{i=1}^{\infty}(-1)^i x_i=$ convergent

2. ,my teacher told me : these 2 conditions are enought but i didnt managed to proof.So would someone confirme the rightness of this?
1. $lim|a_i| \longrightarrow 0$ when $i \longrightarrow \infty$
2. $|a_{i+1} | \le |a_i|$

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Diarmuid · Poincare Conjecture Offline Joined: 25 Aug 2004 Posts: 176

The answer you have for (1) is wrong - the classical counterexample being $a_i=\frac{1}{i}$ , which is decreasing and limits to zero as in the conditions you mention, but the sum of whose terms diverges.

**Posted:** Sun Sep 11, 2005 5:29 am

lomos_lupin

**Posted:** Sun Sep 11, 2005 5:38 am

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Cezar Lupu

it will be $ln 2$ Wink

**Posted:** Sun Sep 11, 2005 5:47 am

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Diarmuid · Poincare Conjecture Offline Joined: 25 Aug 2004 Posts: 176

Well it's not really right for the second one either - e.g. $a_i=\frac{(-1)^i}{i}$ gives $\sum_{i=1}^\infty(-1)^ia_i=\sum_{i=1}^\infty\frac{1}{i}$ , which is the same thing.

If you additionally assume that all the terms are positive (or negative), then you'd have a sufficient condition.

**Posted:** Mon Sep 12, 2005 1:16 am

blahblahblah

1) The only reasonable sufficient condition you're going to get is something like the Cauchy criterion, unless you impose much tighter restrictions on $\{a_i\}$ . In particular, $a_i\to 0$ is not strong enough.

2) If each $a_i$ is positive, then it is enough to have $a_i\geq a_j$ for $i\leq j$ , and $a_i\to 0$ . This is the alternating series test; you may prove it using partial summation.

**Posted:** Mon Sep 12, 2005 1:29 am

perfect_radio

i have a related question: knowing that $\sum_{i=1}^{\infty} |a_i|$ converges, how do you prove that $\sum_{i=1}^{\infty} a_i$ converges?

**Posted:** Fri Sep 16, 2005 10:17 am

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Diarmuid · Poincare Conjecture Offline Joined: 25 Aug 2004 Posts: 176

Triangle inequality on the partial sums?

(i.e. $\left|\sum a_i\right|\le\sum|a_i|$ ).

**Posted:** Fri Sep 16, 2005 11:32 am

perfect_radio

**Posted:** Sun Oct 02, 2005 1:54 pm

_________________
"Germany has offered to send troops to the Lebanon border. I bet Israel's breathing a sigh of relief there. Nothing makes Jewish people feel safer and more secure than the German Army marching on their border."

blahblahblah

Yes, it is enough. Try using the Cauchy criterion.

**Posted:** Sun Oct 02, 2005 2:13 pm