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Math Help - Power Series Problem

  1. #1
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    Power Series Problem

    What is going on here? I understand it somewhat.

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

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

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

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

    Below here makes sense

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

    |x|< 3

    R = 3 ??

    I = (-3,3) - Interval of convergence
    Last edited by Jason76; August 1st 2014 at 08:34 PM.
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  2. #2
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    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).
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  3. #3
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    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 \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:
    \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
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