Can the Alternating Series Test Determine Divergence?

[kidsmoker]

Homework Statement

Find a sequence a_{n} which is non-negative and null but where \sum (-1)^{n+1} a_{n} is divergent.

Homework Equations

Alternating series test:

Let a_{n} be a decreasing sequence of positive real numbers such that a_{n}\rightarrowa as n\rightarrow\infty. Then the series \sum (-1)^{n+1} a_{n} converges.

The Attempt at a Solution

I'm a bit confused by this one. If a_{n} is non-negative and null then it seems like it's decreasing to zero, in which case it satisfies the alternating series test. So how can the sum diverge?!

 

[Dick]

How about (1,0,1/2,0,1/3,0,1/4,0...)? It's null but nondecreasing. The (-1)^(n+1) doesn't help much does it?

 

[kidsmoker]

Ah yeah i see. So you sort of pad it out with zeros to remove the minus terms. Thanks!