Question

Find the sum of this infinite geometric series: 4 - 2.88 + 2.0736- 1.492992+...

Answer 1

Let $r$ be the common ratio between terms. Then

$-2.88=4r\implies r=-0.72$

Now denote the $n$th partial sum of the series by

$S_n=4-2.88+2.0736-1.492992+\cdots+4(-0.72)^{n-1}+4(-0.72)^n$

Multiply both sides by $-0.72$, then subtract from the above to get

$S_n-(-0.72S_n)=4-4(-0.72)^{n+1}$
$1.72S_n=4(1-(-0.72)^{n+1})$
$S_n=2.32558(1-(-0.72)^{n+1})$

As $n\to\infty$, you're left with

$S_\infty=2.32558$