# Can you use the alternating test to conclude if a series is divergent?

• Oct 13th 2012, 06:01 PM
MathIsOhSoHard
Can you use the alternating series test to conclude if a series is divergent?
If an infinite series is not monotonically decreasing and if it does not approach 0 but instead approaches infinity, would this be sufficient information to conclude that the series is divergent since it doesn't meet the alternating test's requirements?
• Oct 13th 2012, 06:36 PM
chiro
Re: Can you use the alternating test to conclude if a series is divergent?
Hey MathIsOhSoHard.

Can you outline the example you are talking about?
• Oct 13th 2012, 06:44 PM
MathIsOhSoHard
Re: Can you use the alternating test to conclude if a series is divergent?
Quote:

Originally Posted by chiro
Hey MathIsOhSoHard.

Can you outline the example you are talking about?

$\sum^\infty_{n=1}(-2)^n\frac{1}{n^2+7}$

Then:
$a_n=\frac{2^n}{n^2+7}$

This does not decrease monotonically nor does it approach zero, so would I be able to conclude that it is divergent based on the alternating series test? Or would the test be inconclusive and I'd have to use another test?
• Oct 13th 2012, 07:02 PM
chiro
Re: Can you use the alternating test to conclude if a series is divergent?
The nth term test should show that this diverges as long as n > 4 so you can break up the series with n = 1 to 4 and then another series with n = 4 onwards and show that absolute value is greater than 1 for all terms which means it diverges.