27 multiple choice 1 point what is the value of ∑(n = 0 to ∞)(-2/3)^n? (a) -2 (b) -2/5 (c) 3/5 (d) 3 (e) the…

Answer:

C. $\frac{3}{5}$

Explanation:

Step1: Identify the series type

The series $\sum_{n = 0}^{\infty}\left(-\frac{2}{3}\right)^n$ is a geometric - series with the general form $\sum_{n=0}^{\infty}ar^n$, where $a = 1$ (when $n = 0$, $\left(-\frac{2}{3}\right)^0=1$) and $r=-\frac{2}{3}$.

Step2: Check the convergence condition

For a geometric series $\sum_{n = 0}^{\infty}ar^n$, it converges if $|r|\lt1$. Here, $\left|-\frac{2}{3}\right|=\frac{2}{3}\lt1$, so the series converges.

Step3: Use the sum formula for a geometric series

The sum formula for a convergent geometric series $\sum_{n = 0}^{\infty}ar^n$ is $S=\frac{a}{1 - r}$. Substituting $a = 1$ and $r=-\frac{2}{3}$ into the formula, we get $S=\frac{1}{1-\left(-\frac{2}{3}\right)}=\frac{1}{1+\frac{2}{3}}=\frac{1}{\frac{3 + 2}{3}}=\frac{3}{5}$.