Q. 4 - Calculus

Question

Which p-series converge and which diverge?

Step-by-Step Solution

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 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.