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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