Question

Determine whether each infinite geometric series diverges or converges. Find the sum if the series converges.

1/2 + 1/16 + 1/128 + ...

Answer 1

\[\frac{1}{2}+\frac{1}{16}+\frac{1}{128}+\cdots=\left(\frac{1}{2}\right)^1+\left(\frac{1}{2}\right)^4+\left(\frac{1}{2}\right)^7+\cdots=\sum_{n=0}^\infty\left(\frac{1}{2}\right)^{1+3n}\]

Answer 2

find the common ratio of the series... if -1 < r < 1 then the series converges

if it does converge the limiting sum is

is \[S_{\infty} = \frac{a}{1 - r}\]

Answer 3

What's the common ratio?

Answer 4

wow its interesting you are doing a geometric series question and don't know what the common ratio is.

you can find it by comparing the ratio of terms


\[\frac{a_{2}}{a_{1}} = \frac{a_{3}}{a_{2}} = ... \]

\[\frac{\frac{1}{16}}{\frac{1}{2}} = \frac{\frac{1}{128}}{\frac{1}{16}}\]

the alternate method is to use

\[a_{n} = a_{1} r^{n -1}\]

you know the 1st term is 1/2 and the 2nd term is 1/16

then

\[\frac{1}{16} = \frac{1}{2} \times r^{(2-1)}\]

solve for r