Summary of Series Tests

The central question in these modules covering series is to determine whether a series converges.  If a series converges, then there are sometimes ways to compute its sum.  If it diverges, then it makes no sense.

We already have some tests.  The following three techniques should be considered first when looking at a series.

1. Geometric series:  When we have $\displaystyle\sum_{n=0}^\infty ar^n$, with $|r|<1$, then the series converges to \frac{a}{1-r}$.
   


2. Telescoping sums:  If our series has telescoping partial sums, then we can determine the convergence/divergence by taking the limit of the few terms in the partial sum (after cancellation), and if it converges we can find its value.


3. Test for divergence:  If $\displaystyle\lim_{n\to\infty}a_n\ne0$, then $\sum a_n$ diverges.  But if $\displaystyle\lim_{n\to\infty}a_n=0$, we need to do more work. 
   

The main techniques that we will cover to determine convergence of series when the steps above don't suffice are in the upcoming sets of modules:

Integral test:  (This module)  Certain infinite sums can be compared to improper integrals.  The sums converge if and only if the integrals converge.

Comparison tests:  If $\sum a_n$ converges and {$b_n$} meets certain criteria as compared to {$a_n$}, then $\sum b_n$ converges.  If $\sum a_n$ diverges and {$b_n$} meets certain criteria as compared to {$a_n$}, then $\sum b_n$ diverges.

Alternating series:  Series whose terms go back and forth between positive and negative have some special properties.  The cancellation between positive and negative terms gives more convergence than you might expect from the size of the terms, and makes convergence of these series easy to determine.

Ratio and Root Tests:  These are the big tests, which often are useful when other tests fail.  In addition, these will be the tests that will allow us to understand radii of convergence for power series.