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Mom's Basement University

Convergence Tests

What You’ll Learn

Section titled “What You’ll Learn”

In this lesson you’ll learn the major convergence tests, which are your toolkit for deciding whether an infinite series adds up to a finite number or not.

The Concept

Section titled “The Concept”

The nth-Term Test (Divergence Test)

Section titled “The nth-Term Test (Divergence Test)”

The simplest check. If the terms don’t approach zero, the series can’t converge

If lim⁡n→∞an≠0, then ∑an diverges\text{If } \lim_{n \to \infty} a_n \neq 0, \text{ then } \sum a_n \text{ diverges}

Important: if the limit is 0, this test tells you nothing. The harmonic series has terms going to 0 but still diverges.

p-Series Test

Section titled “p-Series Test”
∑n=1∞1npconverges if p>1,diverges if p≤1\sum_{n=1}^{\infty} \frac{1}{n^p} \quad \text{converges if } p \gt 1, \quad \text{diverges if } p \leq 1

The bigger the exponent p, the faster the terms shrink, and the easier it is for the sum to stay finite.

The red dots (p = 1, harmonic) keep climbing. The blue dots (p = 2) level off near π²/6. The green dots (p = 3) converge even faster to about 1.202.

Geometric Series Test

Section titled “Geometric Series Test”
∑n=0∞arn=a1−rwhen ∣r∣<1\sum_{n=0}^{\infty} ar^n = \frac{a}{1 - r} \quad \text{when } |r| \lt 1

Comparison Test

Section titled “Comparison Test”

If you have two series with positive terms where each term of one is less than or equal to the corresponding term of the other

  • If the bigger series converges, the smaller one does too
  • If the smaller series diverges, the bigger one does too

Limit Comparison Test

Section titled “Limit Comparison Test”

If two series have positive terms and

lim⁡n→∞anbn=Lwhere 0<L<∞\lim_{n \to \infty} \frac{a_n}{b_n} = L \quad \text{where } 0 \lt L \lt \infty

then both series either converge or both diverge. This is often easier than direct comparison because you only need the limit of the ratio.

Ratio Test

Section titled “Ratio Test”
L=lim⁡n→∞∣an+1an∣L = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|
  • L less than 1: converges absolutely
  • L greater than 1: diverges
  • L equals 1: inconclusive

The ratio test is your go-to for series with factorials or exponentials.

Root Test

Section titled “Root Test”
L=lim⁡n→∞∣an∣nL = \lim_{n \to \infty} \sqrt[n]{|a_n|}

Same conclusions as the ratio test. Useful when the entire term is raised to the nth power.

Worked Examples

Section titled “Worked Examples”

Example 1: p-Series

Determine convergence of the following

∑n=1∞1n2\sum_{n=1}^{\infty} \frac{1}{n^2}

This is a p-series with p = 2. Since 2 is greater than 1, the series converges. (Its sum is π²/6, but the p-series test only tells you it converges, not what it converges to.)

∑n=1∞1n=∑n=1∞1n1/2\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} = \sum_{n=1}^{\infty} \frac{1}{n^{1/2}}

This is a p-series with p = 1/2. Since 1/2 is less than or equal to 1, the series diverges.

Example 2: Ratio Test

Does the series converge or diverge?

∑n=1∞n!3n\sum_{n=1}^{\infty} \frac{n!}{3^n}

Apply the ratio test

L=lim⁡n→∞an+1an=lim⁡n→∞(n+1)!3n+1⋅3nn!L = \lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \lim_{n \to \infty} \frac{(n+1)!}{3^{n+1}} \cdot \frac{3^n}{n!} =lim⁡n→∞(n+1)!⋅3n3n+1⋅n!=lim⁡n→∞n+13=∞= \lim_{n \to \infty} \frac{(n+1)! \cdot 3^n}{3^{n+1} \cdot n!} = \lim_{n \to \infty} \frac{n+1}{3} = \infty

Since L = infinity, which is greater than 1, the series diverges. The factorial grows much faster than the exponential.

Example 3: Comparison Test

Does the following series converge?

∑n=1∞1n2+1\sum_{n=1}^{\infty} \frac{1}{n^2 + 1}

For all n at least 1, we have n² + 1 greater than n², so

1n2+1<1n2\frac{1}{n^2 + 1} \lt \frac{1}{n^2}

We know the p-series with p = 2 converges. Since every term of our series is smaller than the corresponding term of a convergent series, our series converges by the comparison test.

Real-World Application

Section titled “Real-World Application”

Convergence tests matter whenever you use infinite series in practice:

  • Fourier series in audio processing need to converge for the signal reconstruction to work
  • Power series approximations in game engines are only valid within their radius of convergence
  • Numerical methods (like Newton’s method) are analyzed for convergence speed using ratio-like tests
  • Machine learning gradient descent convergence is studied with similar tools

Quiz

Section titled “Quiz”
If $\lim_{n \to \infty} a_n \neq 0$, the series $\sum a_n$
The p-series $\sum 1/n^p$ converges when
The ratio test gives $L = \infty$ for $\sum n!/3^n$, so the series
To show $\sum 1/(n^2 + 1)$ converges, you can compare it to
The ratio test is especially useful for series involving