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Math Help - Find the power series

  1. #1
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    Find the power series

    Find the power series centered at 2 for the function:
     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

    \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:
     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

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

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

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

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

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

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


    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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