Practice Convergence and Divergence in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Quick 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?

Example 1

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

Example 2

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

Example 3

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

Example 4

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