THE P-SERIES - College of Arts and Sciences
[Pages:1]Calculus Notes Grinshpan
THE P-SERIES
The series
1
11
1
np = 1 + 2p + 3p + . . . + np + . . .
n=1
is called the p-series. Its sum is finite for p > 1 and is infinite for p 1.
If p = 1 we have the harmonic series.
For p > 1, the sum of the p-series (the Riemann zeta function (p)) is a monotone decreasing function of p.
For almost all values of p the value of the sum is not known. For instance, the
exact value of the sum
n=1
1 n3
is
a
mystery.
But,
of
course,
one
can
always
find
accurate approximations for any given p.
Some of the known sums and approximations are
1 2 n2 = 6
n=1
1 4 n4 = 90
n=1
1
6
n6 = 945
n=1
1 n3 1.2020569
n=1
1 n5 1.0369278
n=1
1 n7 1.0083493
n=1
One often compares to a p-series when using the Comparison Test.
Example. Test the series
n=1
1 n2 +3
for
convergence.
Solution. Observe that
1
1
n2 + 3 < n2
for every n 1. The series
1 n=1 n2
converges
(p-series with
p = 2 > 1).
So
the
given series converges too, by the Comparison Test.
Or when using the Limit Comparison Test.
Example. Test the series
n=1
n n3/2 +3
for
convergence.
Solution. Observe that
n
1
n3/2
1
: =
=
1 = 0, n .
n3/2 + 3 n n3/2 + 3 1 + 3n-3/2
The series
n=1
1 n
diverges
(p-series
with
p
=
1 2
1).
So
the
given
series
diverges
as well, by the Limit Comparison Test.
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