Question

Which p-series converge and which diverge?

Short answer

The p-test states that:

(i) For $p>1$, the series $\sum _{k=1}^{\infty }\frac{1}{k^p}$converges.

(ii) For $p=1$, the harmonic series $\sum _{k=1}^{\infty }\frac{1}{k}$diverges.

(iii) For $p<0$, the series $\sum _{k=1}^{\infty }\frac{1}{k^p}$diverges.

Step-by-step solution

Step 1. Given Information.

p-series.

Step 2. To determine which p-series converge or diverge.

The p-test states that:

(i) For $p>1$, the series $\sum _{k=1}^{\infty }\frac{1}{k^p}$converges.

(ii) For $p=1$, the harmonic series $\sum _{k=1}^{\infty }\frac{1}{k}$diverges.

(iii) For $p<1$, the series $\sum _{k=1}^{\infty }\frac{1}{k^p}$diverges.