Alternate series question

1. Mar 15, 2005

kdinser

I'm reviewing power series for use in differential equations and I'm having some trouble remembering how to deal with alternating series.

For instance, if I have:
$$\sum(-1)^n(\frac{3}{2})^n$$

if $$a_n=(\frac{3}{2})^n$$
This fails the alternate series test because the limit of $$a_n$$ as n goes to infinity doesn't equal 0.

Can I group the (-1)^n into the fraction and call it a geometric series? In that case, it would diverge, |r| would be greater then 1.

2. Mar 15, 2005

learningphysics

Yes, both ways are fine.

3. Mar 15, 2005

kdinser

thanks for the help.

4. Mar 15, 2005

Data

If the summand doesn't go to zero, the series cannot converge, regardless of whether it is alternating or not (using the most common definition of convergence).

5. Mar 15, 2005

learningphysics

Yes, you're right. This is the best way to solve the problem. The summand doesn't go to zero so the series diverges.