Question

Find a power series representation for the function. Determine the interval of convergence. (Give your power series representation centered at x = 0.) f(x) = 5 2 − x

Answer 1

I'm guessing the function is

$f(x)=\dfrac5{2-x}=5(2-x)^{-1}$

whose first few derivatives are

$\dfrac{\mathrm df}{\mathrm dx}=-5(2-x)^{-2}(-1)=5(2-x)^{-2}$

$\dfrac{\mathrm d^2f}{\mathrm dx^2}=-2\cdot5(2-x)^{-3}(-1)=2!\cdot5(2-x)^{-3}$

with a general pattern of

$\dfrac{\mathrm d^nf}{\mathrm dx^n}=n!\cdot5(2-x)^{-(n+1)}=\dfrac{5n!}{(2-x)^{n+1}}$

Then $f$ has power series (centered at 0)

$\displaystyle\sum_{n=0}^\infty\frac{\frac{5n!}{2^{n+1}}}{n!}x^n=\frac52\sum_{n=0}^\infty\left(\frac x2\right)^n$

which is a geometric series, converging only for

$\left|\dfrac x2\right|$

(this is the interval of convergence)

###

If you know that for $|x|$

$\dfrac1{1-x}=\displaystyle\sum_{n=0}^\infty x^n$

then we can derive the same series by manipulating this result:

$\displaystyle\frac5{2-x}=\frac52\frac1{1-\frac x2}=\frac52\sum_{n=0}^\infty\left(\frac x2\right)^n$