How Can I Systematically Determine Convergence and Divergence of Series?

[tamintl]

Hey there..

Basically I'm struggling with convergence and divergence of series.

I can see if converges and diverges by common sense and thinking through in my head but I struggle to write it down. The definitions in books seem confusing.

Are there any steps I can systematically do every time which will help me determine on paper if the said series div. or conv.?

For example a basic typical question in my course is like the following:

as n ---> infinity

\sum (n^2 - 1) / (n^4 + 1)

Some detailed exlanation in basic language would be greatly appreciated


Regards
Tam

 

[spamiam]

There are many different tools to test the convergence of a series, the most common probably being the comparison test, the integral test, the ratio test, and the nth root test. Here's a link with a big list: http://www.math.hmc.edu/calculus/tutorials/convergence/

In your particular example, I think comparison with the series \sum_{n=1}^\infty \frac{1}{n^2} is the easiest test to use.

 

[paulfr]

Here are some urls with good information on this topic

http://online.math.uh.edu/HoustonACT/
http://online.math.uh.edu/HoustonACT/videocalculus/index.html
http://www.khanacademy.org/#calculus
http://www.stewartcalculus.com/medi...822719&archive=&start_from=&ucat=2&show_cat=2