Calculus Notes
Grinshpan
THE P-SERIES
The series
∞
X
1
n=1
np
=1+
1
1
1
+ p + ... + p + ...
p
2
3
n
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 ζ(p)) is a monotone
decreasing function of p.
For almost all values ofPp the value of the sum is not known. For instance, the
∞
exact value of the sum n=1 n13 is a mystery. But, of course, one can always find
accurate approximations for any given p.
Some of the known sums and approximations are
∞
∞
X
X
π2
1
1
=
≈ 1.2020569
2
n
6
n3
n=1
n=1
∞
X
1
n=1
∞
X
n
=
4
π4
90
1
π6
=
6
n
945
n=1
∞
X
1
n=1
∞
X
n5
≈ 1.0369278
1
≈ 1.0083493
7
n
n=1
One often compares to a p-series when using the Comparison Test.
P∞
Example. Test the series n=1 n21+3 for convergence.
Solution. Observe that
1
1
< 2
2
n
P∞ n + 3
for every n ≥ 1. The series n=1 n12 converges (p-series with p = 2 > 1). So the
given series converges too, by the Comparison Test.
Or when using the Limit Comparison Test.
P∞
Example. Test the series n=1 n3/2n +3 for convergence.
Solution. Observe that
n
1
n3/2
1
√
:
=
=
→ 1 6= 0, n → ∞.
n
n3/2 + 3
n3/2 + 3
1 + 3n−3/2
P∞
The series n=1 √1n diverges (p-series with p = 12 ≤ 1). So the given series diverges
as well, by the Limit Comparison Test.