Question

Find a power series for the function, centered at c. f(x) = 1 9 − x , c = 4 f(x) = [infinity] n = 0 Incorrect: Your answer is incorrect. Determine the interval of convergence. (Enter your answer using interval notation.)

Answer 1

Looks like the given function is

$f(x)=\dfrac1{9-x}$

Recall that for |x| < 1, we have

$\displaystyle\frac1{1-x}=\sum_{n=0}^\infty x^n$

We want the series to be centered around $x=4$, so first we rearrange f(x) :

$\dfrac1{9-x}=\dfrac1{5-(x-4)}=\dfrac15\dfrac1{1-\frac{x-4}5}$

Then

$\dfrac1{9-x}=\displaystyle\frac15\sum_{n=0}^\infty\left(\frac{x-4}5\right)^n$

which converges for |(x - 4)/5| < 1, or -1 < x < 9.