Q. 63 - Calculus

Question

Prove that the geometric sequence ${\left\{r^k\right\}}_{k=0}^{\infty }$ diverges when r < −1.

Step-by-Step Solution

Answer

Geometric sequence ${\left\{r^k\right\}}_{k=0}^{\infty }$ diverges when r<-1

Step 1. Given information

${\left\{r^k\right\}}_{k=0}^{\infty }$ 

Step 2. proof

The given sequence is a geometric sequence and is converging to $\left|L\right|$for r< -1. Therefore, it contradicts the fact that if r>1 then $\left\{r^k\right\}$ diverges as the terms of sequence $\left|{\left\{r^k\right\}}_{k=0}^{\infty }\right|$ are positive and increasing but converging $\left|L\right|$.

Therefore, the assumption is wrong.

Hence, proved.