Chapter 11, Problem 9STP

To determine

Find the option which does not converge to a sum.

Answer to Problem 9STP

Option B is the correct choice.

Explanation of Solution

Given:

Find the option which does not converge to a sum.

4 options are given: $\text{A}:{\displaystyle \sum _{k=1}^{\infty }4·{(\frac{9}{10})}^{k-1}};\text{B}:{\displaystyle \sum _{k=1}^{\infty }\frac{1}{5}·{(\frac{3}{2})}^{k-1}};\text{C}:{\displaystyle \sum _{k=1}^{\infty }\frac{7}{6}·{(\frac{1}{3})}^{k-1}};\text{D}:{\displaystyle \sum _{k=1}^{\infty }(-2)·{(\frac{5}{6})}^{k-1}}$

Concept Used:

Condition for Divergent and Convergent:

If $\left|r\right|<1$ the series is convergent to a sum and if $\left|r\right|>1$ the series is divergent to series

The base of the exponential function is the common ratio r,

Option A, C, and D, the common ratio1s are less than 1, they all are convergent series.

Option B, has the common ratio greater than 1. This series does not converge to a sum, it is a divergent series.

Option B is the correct choice.

Thus, Option B is the correct choice.