P-Series and Ratio Tests

P-Series

Recall that the geometric series is a series where the general term is a constant multiple of rn where r is a constant and n is the index of summation.  We we will look at exponents where the constant is the power and the base is the index.
 Definition of a P-SeriesIf p is a real number then the series         is called a P-Series.

Examples

is a P-series with p  =  -1/2.  Recall that roots are just 1/2 powers and pulling up from the denominator is just changing the sign of the exponent.

is a P-series with p  =  1.  This is just the harmonic series.

is not a P-series since the index is in the exponent.  In fact the above series is a geometric series with r  =  1/2.

P-Series Test

The following test tells us when the p-series converges.

 Theorem:  P-Series Test Consider the series           If p > 1 then the series converges If 0 < p < 1 then the series diverges

Examples

The harmonic series

diverges by the P-series test since

p  =  1  <  1

The series

diverges since

p  =  2  > 1

Ratio Test

If a series converges then the terms must approach zero as n gets large.  However, we have seen that even if the terms approach zero, the series may still diverge.  For example, the harmonic series diverges, however the terms do not approach zero fast enough for convergence.  The next theorem gives us a method that will often detect whether the terms approach zero fast enough.

 Theorem:  The Ratio Test  Let  be a series with nonzero terms.  Then If            then the series converges. If                then the series diverges. If               then try another test.

Remark

The ratio test is especially useful when powers or factorials appear in the general term.  For powers, we have

bn+1

=  b
bn

For factorials, note for example

7!             (7)(6)(5)(4)(3)(2)
=                                      =      7
6!               (6)(5)(4)(3)(2)

In general,

(n+1)!
=  n+1
n!

Example

Determine the convergence or divergence of

We use the Ratio Test:

Hence the series converges by the Ratio Test

Exercises

Determine the convergence or divergence of

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