Can alternating series be grouped as geometric series for convergence?

[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.

 

[learningphysics]

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.

Yes, both ways are fine.

 

[kdinser]

thanks for the help.

 

[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).

 

[learningphysics]

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).

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