Convergence and Divergence Math Example 2

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

Example 2

hard
Determine whether โˆ‘n=1โˆž1n2\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^2} converges using the pp-series test.

Solution

  1. 1
    This is a pp-series โˆ‘1np\sum \frac{1}{n^p} with p=2p = 2.
  2. 2
    The pp-series converges if and only if p>1p > 1.
  3. 3
    Since p=2>1p = 2 > 1, the series converges.
  4. 4
    Its exact sum is ฯ€26\frac{\pi^2}{6} (Basel problem), though the pp-series test only tells us it converges.

Answer

The series converges (p=2>1p = 2 > 1). The sum equals ฯ€26\frac{\pi^2}{6}.
The pp-series test is simple and powerful: โˆ‘1/np\sum 1/n^p converges exactly when p>1p > 1. This includes p=2p=2 but excludes p=1p=1 (harmonic series, which 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 โ†’

More Convergence and Divergence Examples

Example 1 medium

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

Example 3 easy

Does [formula] converge or diverge?

Example 4 medium

Apply the ratio test to [formula].

โ† All Convergence and Divergence Examples Convergence and Divergence Concept