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-Series If p is a real number then the series
is called a P-Series.
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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.
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Theorem: P-Series Test
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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.
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Theorem: The Ratio Test
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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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