Find the sum of the series $\sum_{k=0}^\infty \frac{2^{k+3}}{3^{k}}$

Question

Find the sum of the series $$\sum_{k=0}^\infty \frac{2^{k+3}}{3^{k}}$$

I imagine we are needing to use the geometric series approach (since that's the section I'm studying) but not sure how to get started with an exponent in the numerator and denominator...

Answer 1

Write $$\frac{2^{k+3}}{3^k}=8\times \left(\frac{2}{3}\right)^k$$ and think at a geometric series.

Answer 2

$$ \begin{align}\sum_{k=0}^\infty \frac{2^{k+3}}{3^{k}} & = 8 \sum_{k=0}^\infty \left(\frac23\right)^k \\ & = 8 \times {\frac {1}{1 - \frac23 }} \quad\quad \text { Using the sum of Infinte G.P}\\ &= 8 \times \frac 3{1} =\color{blue}{\boxed{24}}\end{align}$$

Sum of Infinite G.P