How Can the Alternating Series Test Assume N=1?

[georg gill]

Alternating series test:

1. All the $$u_n$$ are all positive

2. $$u_n\geq u_{n+1}$$ for all $$n \geq N$$. For some integer N

3. $$u_n \rightarrow 0$$


I thought it would hold with 2. and that the su m of the N first terms were not $$\infty$$

Here is the theroem just in case:

http://bildr.no/view/1047382

Here they assume N=1 how can they do that?

 

[micromass]

Alternating series test:

1. All the $$u_n$$ are all positive

2. $$u_n\geq u_{n+1}$$ for all $$n \geq N$$. For some integer N

3. $$u_n \rightarrow 0$$


I thought it would hold with 2. and that the su m of the N first terms were not $$\infty$$

How can the sum of the first N terms ever equal infinity?? If you add up finitely many real number, then you never get infinity.

Here is the theroem just in case:

http://bildr.no/view/1047382

Here they assume N=1 how can they do that?

The proof for arbitrary N is very similar. Try to prove it yourself!
(or even better: reduce to the case N=1)