Chapter 12 - Sequences; Induction; the Binomial Theorem - 12.3 Geometric Sequences; Geometric Series - 12.3 Assess Your Understanding - Page 825: 64

Divergent

We are given the geometric series: $\sum_{k=1}^{\infty} 3\left(\dfrac{3}{2}\right)^{k-1}$ Determine the elements of the geometric series: $a_1=3\left(\dfrac{3}{2}\right)^{1-1}=3$ $r=\dfrac{3}{2}$ We compute $|r|$: $|r|=\left|\dfrac{3}{2}\right|=\dfrac{3}{2}$ Because $|r|=\dfrac{3}{2}>1$, the series diverges.