Question

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

Answer 1

what is the common ratio ?

Answer 2

-1/2

Answer 3

yes`

Answer 4

This is an infinite series correct?

Answer 5

yes

Answer 6

is the answer 1/3

Answer 7

Ok, so normally, as you know the sum of a geometric series with common ratio r, and first term \(a_1\) is given by:

\(\large\color{black}{ \displaystyle {\rm S}=\frac{{\rm a}_1(1-{\rm r}^n)}{1-{\rm r}} }\)

Now, for any infinite geometric series (where |r|<1, which is true in this case) \({\rm r}^n\) will be approaching 0, and thus \(1-{\rm r}^n\) will be equivalent to 1-0=1.
\({\rm a}_1~\cdot 1\), is same as just \({\rm a}_1\).

So for an infinite geometric series we therefore write:

\(\large\color{black}{ \displaystyle {\rm S}=\frac{{\rm a}_1}{1-{\rm r}} }\)

Answer 8

\(\large\color{black}{ \displaystyle {\rm S}=\frac{\frac{1}{2}}{1-\left(-\frac{1}{2}\right)} }\)

\(\large\color{black}{ \displaystyle {\rm S}=\frac{\frac{1}{2}}{\frac{2}{2}-\left(-\frac{1}{2}\right)} }\)

\(\large\color{black}{ \displaystyle {\rm S}=\frac{\frac{1}{2}}{\frac{2}{2}+\frac{1}{2}} }\)

\(\large\color{black}{ \displaystyle {\rm S}=\frac{\frac{1}{2}}{\frac{3}{2}} }\)

\(\large\color{black}{ \displaystyle {\rm S}=\frac{1}{3} }\) your answer is correct!!

Answer 9

yay THANKS!!!!!

Answer 10

yw