State the limit of $1/3,1/9,1/27,1/81,1/243....(1/3)^n$ ?

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#1/3,1/9,1/27,1/81,1/243....(1/3)^n#

How do you state the limit of this sequence?

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Answer 1 · 2018-03-26T17:15:14

$3/2$

The problem can be rewritten as
$\sum_{n=0}^{oo}(1/3)^n$

And we can see that is a Geometric series. To understand wheather the series converges or not we have to solve the following inequality
$|1/3|<1$

In this case it's easy to say that is alway true
$1/3<1$

So we can finally state that the series converges to the following value
$1/(1-q)$ where $q=1/3$

$1/(1-1/3)$

$1/(2/3)$

and the value of convergence is
$3/2$