Question

find the sum of the following infinite geometric series, if it exists.
1/2 + (-1/4)+(1/8)+(-1/16)+...

Answer 1

\[\frac{\frac 12 }{1+\frac 12}= \frac 13 \]

Answer 2

Try first by grouping terms as follows.
\[\left( \frac{1}{2}-\frac{1}{4}\right) + \left(\frac{1}{8}-\frac{1}{16}\right)+\cdots=\frac{1}{4}+\frac{1}{16}+\cdots=\sum_{n=1}^\infty \left(\frac{1}{4} \right)^n\]This is now just a simple geometric series.

Answer 3

find the common ratio by comparing the terms

\[\frac{T _{2}}{T _{1}} = \frac{T _{3}}{T2} = \frac{T _{4}}{T _{3}}...\]

if |r|<1 use

\[s=\frac{a}{1-r}\] a = 1st term r = common ratio

Answer 4

First Number a: 1/2
Ratio: -1/2
Formula of Finding Sum of Infinite Geometric Series when |r|<1 is
\[{a}/{1-r}\]

(1/2)/(3/2)=1/3

Answer 5

okay thanks I think i get it :)