The geometric series

is convergent if and its sum is

If    , the geometric series is divergent.

EXAMPLE 7: Find the sum of the geometric series.

The first term is   and the common ratio is  . Since  ,the series is convergent and its sum is  

EXAMPLE 8: Is the series   convergent or divergent?

Let’s rewrite the nth term of the series in the form 

We recognize this series as a geometric series with  and . Since , the series diverges

EXAMPLE 9: Show that the harmonic series

is divergent.

For this particular series it’s convenient to consider the partial sums S2, S4,S8, S16, S32, . . . and show that they become large. 

Similarly,  , and in general

This shows that   and so   is divergent. Therefore, the harmonic

series diverges.

Theorem If the series  is convergent, then   .