Answer
The sequence $b,0,{b^2},0,{b^3},0,{b^4},...$ converges for $0 \lt b \lt 1$.
Work Step by Step
The sequence is oscillating between $0$ and ${b^n}$, that is, the general term is given by
${a_n} = \left\{ {\begin{array}{*{20}{c}}
{{b^n},}&{{\rm{if}}}&n&{{\rm{is}}}&{{\rm{odd}}}\\
{0,}&{{\rm{if}}}&n&{{\rm{is}}}&{{\rm{even}}}
\end{array}} \right.$
Thus, the sequence diverges unless $\mathop {\lim }\limits_{n \to \infty } {b^n} = 0$.
Recall that if $b \gt 1$, then ${b^x} \to \infty $ as $x \to \infty $, and if $0 \lt b \lt 1$, then ${b^x} \to 0$ as $x \to \infty $. Therefore, for $0 \lt b \lt 1$ the sequence $b,0,{b^2},0,{b^3},0,{b^4},...$ converges. It converges to $0$.
