Calculus: P-Series
Definition of P-Series
A P-Series is a series of the form:
∞
X 1
n=1
np
where p is a positive constant.
Convergence and Divergence
• Converges if p > 1.
• Diverges if p ≤ 1.
Why?
The intuition is that when p > 1, the terms shrink quickly enough to make the infinite sum
finite. When p ≤ 1, the terms decrease too slowly, so the sum grows without bound.
Harmonic Series
The harmonic series is the special case when p = 1:
∞
X 1
n=1
n
This series diverges, even though the terms go to zero.
Comparison with Integral Test
The convergence of P-Series can be confirmed using the Integral Test:
Z ∞
1
dx
1 xp
1
Definition of P-Series
A P-Series is a series of the form:
∞
X 1
n=1
np
where p is a positive constant.
Convergence and Divergence
• Converges if p > 1.
• Diverges if p ≤ 1.
Why?
The intuition is that when p > 1, the terms shrink quickly enough to make the infinite sum
finite. When p ≤ 1, the terms decrease too slowly, so the sum grows without bound.
Harmonic Series
The harmonic series is the special case when p = 1:
∞
X 1
n=1
n
This series diverges, even though the terms go to zero.
Comparison with Integral Test
The convergence of P-Series can be confirmed using the Integral Test:
Z ∞
1
dx
1 xp
1
