Convergence and Divergence Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Convergence and Divergence.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A series converges if the sequence of its partial sums approaches a finite limit. A series diverges if the partial sums grow without bound or oscillate without settling.

Convergence means the infinite sum adds up to a finite number—each new term adds less and less, and the total stabilizes. Divergence means the sum either blows up to infinity or never settles down. The key question: does adding infinitely many terms produce a finite result?

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The terms going to zero is necessary but NOT sufficient for convergence. The harmonic series proves this: \frac{1}{n} \to 0 but \sum \frac{1}{n} = \infty. To determine convergence, you need specific tests (comparison, ratio, integral test, etc.).

Common stuck point: The divergence test only goes one way: if a_n \not\to 0, the series diverges. But a_n \to 0 does NOT mean convergence. You need a stronger test to prove convergence.

Sense of Study hint: Start with the divergence test, then try comparing your series to a known p-series or geometric series.

Worked Examples

Example 1

medium
Use the ratio test to determine whether \displaystyle\sum_{n=1}^{\infty} \frac{n}{2^n} converges or diverges.

Solution

  1. 1
    a_n = \frac{n}{2^n}. Compute L = \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|.
  2. 2
    \frac{a_{n+1}}{a_n} = \frac{(n+1)/2^{n+1}}{n/2^n} = \frac{n+1}{n} \cdot \frac{1}{2}.
  3. 3
    \lim_{n\to\infty} \frac{n+1}{2n} = \frac{1}{2}.
  4. 4
    Since L = \frac{1}{2} < 1, the series converges absolutely.

Answer

The series converges (ratio test: L = \frac{1}{2} < 1).
The ratio test compares consecutive term sizes. L < 1 means terms shrink geometrically fast enough for the sum to be finite. The series sum is actually 2 (computed via differentiation of the geometric series).

Example 2

hard
Determine whether \displaystyle\sum_{n=1}^{\infty} \frac{1}{n^2} converges using the p-series test.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Does \displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{1/2}} converge or diverge?

Example 2

medium
Apply the ratio test to \displaystyle\sum_{n=0}^{\infty} \frac{3^n}{n!}.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

serieslimitinfinite geometric series