Question

Geometric Series and p-series:

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

For r_____, the geometric series$\sum _{k=0}^{\infty }{r}^k$converges to the sum_____.

Short answer

The required answer is For $\left|r\right|<1$, the geometric series$\sum _{k=0}^{\infty }{r}^k$converges to the sum$\frac{1}{1-r}$

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 $\left|r\right|<1$, the geometric series $\sum _{k=0}^{\infty }{r}^k$ converges to the sum$\frac{1}{1-r}$