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Thread: Find the power series

  1. #1
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    Find the power series

    Find the power series centered at 2 for the function:
    $\displaystyle F(x)= \frac{3}{5-x}$
    so there are a lot of steps that i wrote out because i was trying to solve it using taylor series, so i dont want to write them all out but ill just show what i got at the end. But i get something really weird and i dont even think it makes sense

    $\displaystyle \frac{3n!n!}{3^(n+1)}(x-2)^n$

    ** can you even have n! x n! ?? ... im just really confused
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  2. #2
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    Quote Originally Posted by Tascja View Post
    Find the power series centered at 2 for the function:
    $\displaystyle F(x)= \frac{3}{5-x}$
    so there are a lot of steps that i wrote out because i was trying to solve it using taylor series, so i dont want to write them all out but ill just show what i got at the end. But i get something really weird and i dont even think it makes sense

    $\displaystyle \frac{3n!n!}{3^(n+1)}(x-2)^n$

    ** can you even have n! x n! ?? ... im just really confused
    $\displaystyle f(x) = \frac{3}{5-x}$ ... $\displaystyle f(2) = 1$

    $\displaystyle f'(x) = \frac{3}{(5-x)^2}$ ... $\displaystyle f'(2) = \frac{1}{3}$

    $\displaystyle f''(x) = \frac{6}{(5-x)^3}$ ... $\displaystyle f''(2) = \frac{2}{3^2}$

    $\displaystyle f'''(x) = \frac{18}{(5-x)^4}$ ... $\displaystyle f'''(2) = \frac{2 \cdot 3}{3^3}$

    $\displaystyle f^4(x) = \frac{72}{(5-x)^5}$ ... $\displaystyle f^4(2) = \frac{2 \cdot 3 \cdot 4}{3^4}$


    $\displaystyle f(x) = 1 + \frac{x-2}{3} + \frac{(x-2)^2}{3^2} + \frac{(x-2)^3}{3^3}+ ... + \frac{(x-2)^n}{3^n}+ ...$
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