What is the sum of the infinite geometric series 2+2/3+2/9+2/27+...?

Question

What is the sum of the infinite geometric series 2+2/3+2/9+2/27+...?

1 Answer

Impact of this question

20998 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-06-22T00:18:17

$3$ $3$

Explanation:

a geometric series can be defined as: $a_{n} = a_{1} \cdot {(r)}^{n - 1}$ $a_n = a_1*(r)^(n-1)$ where $a_{1}$ $a_1$ is the first value of the series and $r$ $r$ is the common ratio.

The common ratio is $\frac{1}{3}$ $1/3$ and the first term is $2$ $2$ so our series is:

$a_{n} = 2 \cdot {(\frac{1}{3})}^{n - 1}$ $a_n = 2*(1/3)^(n-1)$

You can only sum an infinite geometric series if it is convergent, that is, that it converges to one value. A series is convergent if $| r | < 1$ $abs(r) < 1$ which in this case it is.

The equation to sum an infinite series is: $\frac{a_{1}}{1 - r}$ $a_1/(1-r)$ so by plugging all values we get: $\frac{2}{1 - \frac{1}{3}}$ $2/(1-1/3)$ which is $\frac{2}{\frac{2}{3}}$ $2/(2/3)$ or $\frac{6}{2}$ $6/2$ which simplifies to $3$ $3$

Science

Math

Humanities

... and beyond

What is the sum of the infinite geometric series 2+2/3+2/9+2/27+...?

1 Answer

Explanation:

Related questions

Impact of this question