1. Power Series Problem

What is going on here? I understand it somewhat.

Find the interval of convergence and power series representation for the function.

$\displaystyle f(x) = \dfrac{2}{3 - x} = \dfrac{2}{3} (\dfrac{1 }{1 - (x/3) }$

$\displaystyle = (\dfrac{2}{3})$Summation sign goes here - n = 0 to infinity$\displaystyle (\dfrac{x}{3})^{n}$ ??? - What is the meaning of the n power over $\displaystyle \dfrac{x}{3}$?

$\displaystyle = (2)$Summation sign goes here - n = 0 to infinity$\displaystyle \dfrac{1}{3^{n+1}} x^{n}$ ??? - What is the meaning of n and n + 1 powers here?

Below here makes sense

$\displaystyle |\dfrac{x}{3}|< 1$

$\displaystyle |x|< 3$

$\displaystyle R = 3$ ??

$\displaystyle I = (-3,3)$ - Interval of convergence

2. Re: Power Series Problem

Again, what I said in the other post.
If you have read that post first, (2/3) * sum (n=0~inf.) (x/3)^n should make sense to you.
Now, time for some algebraic manipulation: anything other than n can enter or exit the sigma sign FREELY.
(2/3) * sum (n=0~inf.) (x/3)^n = (2/3) * sum (n=0~inf.) x^n/3^n = 2 * sum (n=0~inf.) x^n/3^n * (1/3) = 2* sum (n=0~inf.) x^n/3^(n+1).

3. Re: Power Series Problem

This is puzzling. If you now the term "power series" then you should know that it is an infinite series of the form $\displaystyle \sum_{n=0}^\infty a_nx^n$. As to "What is the meaning of the n power over x/3?" it means exactly what it always has: the term x/3 multiplied by itself n times:
$\displaystyle \sum_{n=0}^\infty \left\frac{x}{3}\right)^n= 1+ frac{x}{3}+ \left(\frac{x}{3}\right)^3+ \left(\frac{x}{3}\right)^3+ \cdot\cdot\cdot= 1+ \frac{x}{3}+ \frac{x^2}{9}+ \frac{x^3}{27}+ \cdot\cdot\cdot$