Divergent Test for a Series

Theorem If  does not exist or if  , then the series   

EXAMPLE 10:  Show that the series    diverges.

So the series diverges by the Test for Divergence.

Note If we find that   , we know that   is divergent. If we find

that  , we know nothing about the convergence or divergence of   .

Remember if   , the series o an might converge or it might diverge.

Theorem If   and    are convergent series, then so are the series  (where c is a constant),

EXAMPLE 11: Find the sum of the series  

The series   is a geometric series with     and   , so 

Note A finite number of terms doesn’t affect the convergence or divergence of a series. For instance, suppose that we were able to show that the series