Question

Find the sum of the following infinite geometric series, if it exists. ½ + -¼ + ⅛ + -1/16 +...

Answer 1

Hmm

each term is multiplied by a constant
1/2 times what=-1/4? that is -1/2
-1/4 times what=1/8? that is -1/2

so r=-1/2

does it converge or diverge?
if it converges, it has an infinite sum
if it diverges, it does not

if |r|<1 then it converges
|-1/2|<1?
1/2<1?
true
it converges and has a sum

the sum of an infinite geometric series is
$S_n=\frac{a_1}{1-r}$ where a1 is the first term and r=common ratio

a1=1/2
r=-1/2

$S_{\infty}=\frac{\frac{1}{2}}{1-(\frac{-1}{2})}$
$S_{\infty}=\frac{\frac{1}{2}}{1+\frac{1}{2}}$
$S_{\infty}=\frac{\frac{1}{2}}{\frac{3}{2}}$
$S_{\infty}=(\frac{1}{2})(\frac{2}{3})$
$S_{\infty}=\frac{2}{6}$
$S_{\infty}=\frac{1}{3}$



the sum is $\frac{1}{3}$