Science:Infinite Series Module/Units/Unit 2/2.2 The Integral Test/2.2.07 The p-series

Problem

For what values of p does the infinite series

$\sum _{k=1}^{\infty }{\frac {1}{k^{p}}}$

converge?

Complete Solution

Step (1): Consider p > 0 and p ≠ 1

When p > 0 and p ≠ 1, the function

$f(x)={\frac {1}{x^{p}}}$

is continuous, decreasing, and positive when x is in the interval [1,∞). Using the integral test,

${\begin{aligned}\int _{1}^{\infty }{\frac {1}{x^{p}}}dx&=\lim _{b\rightarrow \infty }\int _{1}^{b}{\frac {1}{x^{p}}}dx\\&=\lim _{b\rightarrow \infty }{\frac {1}{-(p-1)x^{p-1}}}{\Big |}_{1}^{b}\\&={\begin{cases}\infty &{\text{if }}p<1,\\{\frac {1}{p-1}}&{\text{if }}p>1\end{cases}}\end{aligned}}$

Therefore, the infinite series converges when p > 1, and diverges when p is in the interval (0,1).

Step (2): Consider p ≤ 0 and p = 1

If p=1, then we have the harmonic series

$\sum _{k=1}^{\infty }{\frac {1}{k}}$

which we know diverges.

If p ≤ 0, the infinite series diverges (by the divergence test).

Therefore, the given series only converges for p > 1.

The p-Series

The result of this example can be summarized as follows.

The p-Series
The p-series $\sum _{k=1}^{\infty }{\frac {1}{k^{p}}}$ is convergent if p > 1 and divergent if p ≤ 1.

Much like a geometric series, we can use this result to determine whether a given infinite series converges by inspection. For example, the infinite series

$\sum _{k=1}^{\infty }{\frac {1}{\sqrt {1+k}}}$

diverges because it is a p-series with p equal to 1/2 (you may want to let u=(1+k) to see this).