Question

Geometric Series and p-series:

Suppose r is a nonzero real number and p > 0. Fill in the blanks.

For p______ , the p-series $\sum _{k=0}^{\infty }\frac{1}{k^p}$converges and for p______, the series diverges.

Short answer

The required answer is for p>1 , the p-series $\sum _{k=0}^{\infty }\frac{1}{k^p}$converges and for p<1, the series diverges.

Step-by-step solution

Step 1. Given Information  

The given data isr is a nonzero real number andp > 0

Step 2. Explanation  

By using the concept of convergence of a geometric series,

Suppose r is a non zero real number. Then,

For p>1 , the p-series $\sum _{k=0}^{\infty }\frac{1}{k^p}$converges and for p<1, the series diverges.