infinite series (idea)

 inf
---         1
\         ---- 
/___      ln(n)
 n=2

Diverges, even though the lim_n->inf1/ln(n) = 0...

nth-Term Test for Divergence

If a sequence {a_n} does not converge to 0, then the series Sigma(a_n) diverges. This does not imply that if the sequence {a_n} converges to 0 that the series Sigma(a_n) converges; the test is merely inconclusive. In other words...

 inf
---         n
\         ---- 
/___        2
 n=1

Diverges because lim_n->inf n/2 = infinity, which does not exist.

Geometric Series

Geometric series are rather simple when it comes to determining their convergence or divergence. First, it would be best to define geometric series.

 inf
---         
\            ar^n = a + ar + ar^2 + ar^3 +... ar^n +..., r != 0
/___         
 n=0

The r term is referred to as the ratio. If 0 < |r| < 1, then the geometric series converges. To determine the series, if it converges, simply follow the formula S = a/(1-r).

Integral Test

This is another rather simple test that works wonderfully for all easily integrable problems. If f is a positive, continuous, decreasing function for n >= 1, and if f(n) = a_n, then both

 |\ infinity                      
 |   f(x) dx    and               
\| 1           
 
 inf
---         
\      an
/___         
 n=0

either converge or diverge.

P-Series

P-series are are special series defined as follows:

 inf
---    
\         1/n^p = 1/1^p + 1/2^p +... 1/n^p
/___   
 n=1

To figure out the divergence or convergence of these series, just look at the p. If 0 < p <= 1, then the series diverges. If p > 1, the series converges.

Comparison Tests

If a_n <= c_n for all n, and

 inf                                                                            
---                                                                        
\          cn converges, then        
/___                                                                        
 n=1             
                                                          
 inf                                                                            
---                                                                        
\          an converges        
/___                                                                        
 n=1  

And if c_n <= a_n,

 inf                                                                         
---                                                                        
\            cn diverges, then           
/___                                                                         
 n=1            
                   
 inf                                                                         
---                                                                        
\          an diverges
/___                                                                         
 n=1                               

In a similar light, suppose a_n > 0 and d_n > 0. lim_n-> d_n/a_n = L, where L is finite and positive, then the series

inf                                                       
---                                                    
\            dn and         
/___                                                    
 n=1        

 inf                                                       
---                                                    
\            an       
/___                                                    
 n=1                            

both either converge or diverge.

There are several other tests for divergence and convergence, but the preceeding tests are a solid foundation for dealing with infinite series.