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I have a question that I am kind of stuck on. I am good with series stuff, but this one kind of threw me off

$$\sum\limits_{n=1}^{\infty} (-9)^nx^n. $$

This is the question and I was supposed to find out the values of where x converges, and I found that out to be from $\frac{-1}{9} < x < \frac{1}{9}$

How can I find the sum of the series for those values of $x$?

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This is the infinite geometric series $\sum_1^\infty (-9x)^n$. You have probably known this sum for a long time. – André Nicolas Mar 30 '13 at 3:25
yeah but how do i find the sum for those values of x. its not infinty for some reason – MathGeek Mar 30 '13 at 3:27
When $|r|\lt 1$, $a+ar+ar^2+\cdots =\frac{a}{1-r}$. In our case $a=-9x$, $r=-9x$. – André Nicolas Mar 30 '13 at 3:35
up vote 5 down vote accepted

Given the closed form of the geometric series: $$ \sum_{n=1}^\infty a^n = \frac{a}{1-a} $$

Plug $a = -9x$ to get: $$ \sum_{n=1}^\infty (-9)^nx^n = \frac{-9x}{1+9x} $$

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i dont get it, so is the sum 0? – MathGeek Mar 30 '13 at 3:06
@MathGeek The sum depends on $x$. For example if $x = 1/18$ then the sum is $-1/3$. – Ayman Hourieh Mar 30 '13 at 3:07
but i know my series only converges from -1/9 to 1/9 is my x 1/9 then ? :s – MathGeek Mar 30 '13 at 3:08
@MathGeek $x$ can take any value in $(-1/9, 1/9)$. The sum is a function of $x$. You cannot give a fixed number as the sum for all values of $x$. – Ayman Hourieh Mar 30 '13 at 3:11
if i integrate from -1/9 to 1/9 i get it does not exist :s – MathGeek Mar 30 '13 at 3:19

To compute the interval of convergence for a power series $\sum a^n x^n$, compute $$\lim_{n\to \infty} \sqrt[\large n]{|a_n|} = \lim_{n\to\infty} \sqrt[\large n]{9^n}$$

In this case, that gives us $9$, so the radius of convergence is $\dfrac {1}9$ which will converge when $|x| \lt \dfrac 19$

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yes i understand that i got the answer that it only converges from x betweeen -1/9 t0 1/9 but i dont get the next question "Find the sum of the series for those values of x" – MathGeek Mar 30 '13 at 3:16
I think you mean the interval of convergence is $|x|<1/9$. – Alex Becker Mar 30 '13 at 3:17
but how do i find the sum of the series now thats my main question. i know how to get the interval of convergence – MathGeek Mar 30 '13 at 3:18
Perhaps you need to take twice the integral from $0$ to $1/9$ of $$\frac{-9x}{1 + 9x}$$, which is $\dfrac 29 (\ln(2) - 1)$ – amWhy Mar 30 '13 at 3:34

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