Question

Find the sum of the series. 1 + z 3 + z2 9 + z3 27 + $$ correct: your answer is correct. for what values of the variable does the series converge to this sum? (enter your answer using interval notation.)

Answer 1

$1+\dfrac z3+\dfrac{z^2}9+\dfrac{z^3}{27}+\cdots=\displaystyle\sum_{n=0}^\infty\left(\frac z3\right)^n$

which converges for $\left|\dfrac z3\right|$ or $|z|$ to $\dfrac1{1-\frac z3}=\dfrac3{3-z}$.