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
[tex]S_n=\frac{a_1}{1-r}[/tex] where a1 is the first term and r=common ratio
a1=1/2
r=-1/2
[tex]S_{\infty}=\frac{\frac{1}{2}}{1-(\frac{-1}{2})}[/tex]
[tex]S_{\infty}=\frac{\frac{1}{2}}{1+\frac{1}{2}}[/tex]
[tex]S_{\infty}=\frac{\frac{1}{2}}{\frac{3}{2}}[/tex]
[tex]S_{\infty}=(\frac{1}{2})(\frac{2}{3})[/tex]
[tex]S_{\infty}=\frac{2}{6}[/tex]
[tex]S_{\infty}=\frac{1}{3}[/tex]
the sum is [tex]\frac{1}{3}[/tex]
