Q. 11 - Calculus

Question

Explain why all the terms of a divergent geometric series are nonzero.

Step-by-Step Solution

Answer

The terms of a divergent geometric series are nonzero because if any one of the term is zero, then the series will be convergent.

Step 1. Given information

If the common ratio of a geometric progression is greater then one then the series is divergent geometric series.

Step 2. Explanation

Consider the divergent geometric series $\sum _{k=1}^{\infty }c{r}^k, r>1$.

The geometric series has zero terms if either $c=0, or r=0$.

If any one of them is zero then the product $c{r}^k$ is zero.

Step 3. Convergent and divergent geometric series

The product $c{r}^k$ is zero, and then the geometric series $\sum _{k=1}^{\infty }c{r}^k$ has the partial sum equal to zero.

The sequence of partial sum of the series is zero.

The series with zero is constant and is convergent.

Therefore, for divergent geometric series both c and r should be non-zero.