Functions expressed as power series

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  1. Expressing Functions as Power Series
    Express the following functions as power series, and then find the interval of convergence:
    1. 11x2 \frac{1}{1-x^2}
    2. 12x \frac{1}{2-x}
    3. x21+3x2 \frac{x^2}{1+3x^2}
  2. Irregular Function expressed as a Power Series
    Express the function f(x)=ln(7x)f(x)=ln(7-x) as a power series and find the interval of convergence.
    1. Derivative of a Function expressed as a Power Series
      Express the derivative of this function f(x)=11xf(x)=\frac{1}{1-x} as a power series and find the interval of convergence.
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      Topic Notes
      In this section, we will learn how to convert functions into power series. Most of the functions we will be dealing with will be converted into a geometric series. After converting them into a power series, we will find the interval of convergence. Keep in mind that we do not have to check the endpoints of the inequality because we automatically know they will be divergent. Afterwards, we will look at an irregular function and express it as a power series. Integrals will be involved here. We will also take the derivative of a function and express that as a power series.
      Note *A formula that may be of use when expressing functions into power series:

      11r=n=0rn\frac{1}{1-r}=\sum_{n=0}^{\infty}r^n knowing that 1-1 < rr < 11

      When finding the interval of convergence, there is no need to check the endpoints. This is because the sum of the geometric series strictly converges only when 1-1 < rr < 11, and not at r=1r=1.

      If the function f(x)f(x) has a radius of convergence of RR, then the derivative and the anti-derivative of f(x)f(x) also has a radius of convergence of RR.