I have the function f(x) = $\dfrac{2x}{\left(x-5\right)^2}$, and I'm supposed to "find the first five non-zero terms of power series representation centered at x = 0."
Using $\frac{f^{(n)}(0)}{n!}$, the first non-zero terms I get are:
C1 = 2/25
C2 = 4/125
C3 = 6/625
C4 = 8/3125
C5 = 2/3125
However, WebWork marks these as incorrect. Any alternative methods or corrections to the work above are gladly appreciated.
On
$\begin{array}\\ \dfrac{2x}{\left(x-5\right)^2} &=\dfrac{2x}{x^2\left(1-5/x\right)^2}\\ &=\dfrac{2}{x}\sum_{n=0}^{\infty} (n+1)(5/x)^n\\ &=\dfrac{2}{x}(1+2\dfrac{5}{x}+3(\dfrac{5}{x})^2+4(\dfrac{5}{x})^3+5(\dfrac{5}{x})^4+...)\\ &=\dfrac{2}{x}(1+\dfrac{10}{x}+\dfrac{75}{x^2}+\dfrac{500}{x^3}+\dfrac{3125}{x^4}+...)\\ &=\dfrac{2}{x}+\dfrac{20}{x^2}+\dfrac{150}{x^3}+\dfrac{1000}{x^4}+\dfrac{6250}{x^5}+...)\\ \end{array} $
On
Remember, this an online system. Each of your coefficients is correct but perhaps you are missing the variables. For instance, the first term is $2x/25$ not its coefficients (because there can be constant terms in a Taylor Series). Similarly, the second term is $4x^2/125$, and so forth.
It could also be because it wants the coefficients of the terms but in order of powers, so it would $0$, $2/25$, $4/125$, etc. Though this seems less like than the first.
Overall, this seems less an issue of the mathematics - as your terms are correct, as it does how you are entering it or with the system itself. Just send a quick email to your instructor! They can see the solutions and thus perhaps more quickly tell what is incorrect about how the system wants the answer. As an instructor, I always found WebWork/WebAssign to be a bit wonky from time to time.
Hint:
You don't have to compute the successive derivatives. Instead rewrite the fraction as $$\frac{2x}{\left(x-5\right)^2}=\frac{2x}{25}\frac 1{\Bigl(1-\cfrac x5\Bigr)^2}=\frac{2x}5\left(\frac 1{1-\smash[b]{\cfrac x5}}\right)'$$ and use the power series expansion $\;\dfrac1{1-u}=1+u+u^2+\dotsm$