Series Tests 

Given a series sum(a[n], n = 1 .. infinity) , you may use these tests to determine whether or not the series converges or diverges.  Be sure to go through this list in order because they are ranked from easiest to hardest: 

1. Divergence Test - If limit(a[n], n = infinity) <> 0 or DNE, then the series sum(a[n], n = 1 .. infinity) diverges.   

                                  If limit(a[n], n = infinity) = 0, then we don't know anything about the series. 

2. Harmonic Series - sum(1/n, n = 1 .. infinity) diverges. 

3. Geometric Series - for -1< r < 1, diverges for r < -1 or r > 1. 

 

Note:  If you write out the first few terms of the series, the first term is "a" and the second term is a*r.

 

4. Alternating Series Test - For sum((-1)^n*b[n], n = 1 .. infinity), if both of the following conditions are satisfied: 

            i) limit(b[n], n = infinity) = 0 

             ii) b[n]is decreasing (show < 0, or graph is dec., or  

then sum((-1)^n*b[n], n = 1 .. infinity) is convergent.  If ii) fails then, sum((-1)^n*b[n], n = 1 .. infinity) is divergent by divergence test. 

5. Ratio Test - For if: 

          i) L < 1, then sum(a[n], n = 1 .. infinity)is A.C. (absolutely convergent). 

          ii) L > 1 or DNE, then sum(a[n], n = 1 .. infinity)is D (divergent).    

          iii) 1, then no conclusion.