# Thread: Find the power series

1. ## 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

2. Originally Posted by Tascja
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
$\displaystyle f(x) = \frac{3}{5-x}$ ... $\displaystyle f(2) = 1$

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

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

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

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

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