p-series

Q: What is the difference between a p-series and a geometric series?

A p-series has the form $\frac{1}{n^p}$ where the variable $n$ is in the base of the denominator. A geometric series has the form $ar^n$ where the variable $n$ is in the exponent. A geometric series converges when $|r| 1$. They are tested with entirely different convergence criteria.

Q: Why does the p-series diverge when p = 1?

When $p = 1$, the p-series becomes the harmonic series $\sum \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \cdots$, which diverges. Even though the individual terms approach zero, they decrease too slowly for the partial sums to remain bounded. This can be proven using the integral test: $\int_1^{\infty} \frac{1}{x}\,dx = \ln x \big|_1^{\infty} = \infty$.

Q: How do you prove the p-series convergence rule?

The standard proof uses the integral test. You compare $\sum \frac{1}{n^p}$ with the improper integral $\int_1^{\infty} \frac{1}{x^p}\,dx$. When $p > 1$, this integral equals $\frac{1}{p-1}$, which is finite, so the series converges. When $p \leq 1$, the integral diverges, so the series diverges.

p-series

A series of the form 1/1^p + 1/2^p + 1/3^p + ... + 1/n^p + ... or Sum from n=1 to infinity of 1/n^p , where p > 0. Often employed when using the comparison test and the limit comparison test.

Note: The harmonic series is a p-series with p =1.

p-series formula: sum from n=1 to infinity of 1/n^p. Diverges if 0 < p ≤ 1; converges if p > 1.

Key Formula

\sum_{n=1}^{\infty} \frac{1}{n^p} = \frac{1}{1^p} + \frac{1}{2^p} + \frac{1}{3^p} + \cdots

Where:

$n$ = The index of summation, starting at 1 and increasing through all positive integers
$p$ = A positive constant that determines convergence: the series converges if p > 1 and diverges if 0 < p ≤ 1

Worked Example

Problem: Determine whether the p-series

\sum_{n=1}^{\infty} \frac{1}{n^3}

converges or diverges.

Step 1: Identify the series as a p-series and find the value of p.

\sum_{n=1}^{\infty} \frac{1}{n^3} \implies p = 3

Step 2: Apply the p-series convergence rule: a p-series converges if and only if p > 1.

p = 3 > 1

Step 3: Since p = 3 is greater than 1, the condition for convergence is satisfied.

\sum_{n=1}^{\infty} \frac{1}{n^3} \text{ converges}

Answer: The series

\sum_{n=1}^{\infty} \frac{1}{n^3}

converges because

p = 3 > 1

Another Example

This example shows how a p-series is commonly used as a benchmark in the comparison test rather than being tested directly.

Problem: Use the p-series test and the comparison test to determine whether

\sum_{n=1}^{\infty} \frac{1}{n^2 + 5}

converges or diverges.

Step 1: Notice that this is NOT itself a p-series because the denominator is

n^2 + 5

, not

n^p

. However, for large n it behaves like

\frac{1}{n^2}

, which is a p-series with

p = 2

\frac{1}{n^2 + 5} \quad \text{vs.} \quad \frac{1}{n^2}

Step 2: Since

n^2 + 5 > n^2

for all

n \geq 1

, the given terms are smaller than the corresponding p-series terms.

0 < \frac{1}{n^2 + 5} < \frac{1}{n^2} \quad \text{for all } n \geq 1

Step 3: Confirm that the comparison series

\sum \frac{1}{n^2}

converges by the p-series test (

p = 2 > 1

\sum_{n=1}^{\infty} \frac{1}{n^2} \text{ converges } (p = 2 > 1)

Step 4: By the direct comparison test, since each term of our series is positive and less than the corresponding term of a convergent series, our series also converges.

\sum_{n=1}^{\infty} \frac{1}{n^2 + 5} \text{ converges by comparison}

Answer: The series

\sum_{n=1}^{\infty} \frac{1}{n^2 + 5}

converges by the comparison test with the convergent p-series

\sum \frac{1}{n^2}

Frequently Asked Questions

What is the difference between a p-series and a geometric series?

A p-series has the form

\frac{1}{n^p}

where the variable

n

is in the base of the denominator. A geometric series has the form

ar^n

where the variable

n

is in the exponent. A geometric series converges when

|r| < 1

, while a p-series converges when

p > 1

. They are tested with entirely different convergence criteria.

Why does the p-series diverge when p = 1?

When

p = 1

, the p-series becomes the harmonic series

\sum \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \cdots

, which diverges. Even though the individual terms approach zero, they decrease too slowly for the partial sums to remain bounded. This can be proven using the integral test:

\int_1^{\infty} \frac{1}{x}\,dx = \ln x \big|_1^{\infty} = \infty

How do you prove the p-series convergence rule?

The standard proof uses the integral test. You compare

\sum \frac{1}{n^p}

with the improper integral

\int_1^{\infty} \frac{1}{x^p}\,dx

. When

p > 1

, this integral equals

\frac{1}{p-1}

, which is finite, so the series converges. When

p \leq 1

, the integral diverges, so the series diverges.

p-Series vs. Geometric Series

	p-Series	Geometric Series
General form	$\sum_{n=1}^{\infty} \frac{1}{n^p}$	$\sum_{n=0}^{\infty} ar^n$
Variable position	n is in the base of the denominator	n is in the exponent
Convergence condition	Converges when $p > 1$	Converges when $\|r\| < 1$
Exact sum known?	Only for specific values of p (e.g., $p=2$ gives $\pi^2/6$ )	Yes: sum $= \frac{a}{1-r}$
Common use	Benchmark for comparison and limit comparison tests	Direct evaluation of sums; ratio test benchmark

Why It Matters

The p-series is one of the most important benchmark series in calculus. When you need to determine whether an unfamiliar series converges, you will frequently compare it to a p-series using the comparison test or the limit comparison test. Mastering the simple rule — converges if

p > 1

, diverges if

p \leq 1

— gives you a powerful tool for quickly resolving many series convergence problems on AP Calculus and college calculus exams.

Common Mistakes

Mistake: Thinking the p-series converges when p = 1 because the terms

\frac{1}{n}

approach zero.

Correction: Terms approaching zero is necessary but not sufficient for convergence. The harmonic series (

p = 1

) is the classic example of a divergent series whose terms tend to zero. You must have

p > 1

strictly.

Mistake: Confusing a p-series with a geometric series by misidentifying where the variable n appears.

Correction: In a p-series,

n

is the base raised to a constant power:

\frac{1}{n^p}

. In a geometric series, a constant base is raised to the variable power:

r^n

. Check whether the variable is in the base or the exponent before choosing your test.

Related Terms

Series — General concept that p-series is a special case of
Comparison Test — Often uses p-series as the benchmark series
Limit Comparison Test — Frequently pairs unknown series with a p-series
Harmonic Series — The divergent p-series where p = 1
Convergent Series — A p-series is convergent when p > 1
Divergent Series — A p-series is divergent when p ≤ 1
Sigma Notation — Notation used to write p-series compactly