Convergence and Divergence Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

easy
Does ∑n=1∞1n1/2\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{1/2}} converge or diverge?

Solution

  1. 1
    pp-series with p=12<1p = \frac{1}{2} < 1.
  2. 2
    Since p≤1p \leq 1, the series diverges.

Answer

The series diverges.
The pp-series test: ∑1/n1/2\sum 1/n^{1/2} diverges because p=1/2≤1p = 1/2 \leq 1. This is slower than the harmonic series but still diverges.

About Convergence and Divergence

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.

Learn more about Convergence and Divergence →

More Convergence and Divergence Examples

Example 1 medium

Use the ratio test to determine whether [formula] converges or diverges.

Example 2 hard

Determine whether [formula] converges using the [formula]-series test.

Example 4 medium

Apply the ratio test to [formula].

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