Question

(a) Find a power series representation for the function. (Give your power series representation centered at x = 0.) f(x)=x/(10 x**2+1) sum_(n=0)^infinity (-1)**n 10**n x**(2n+1) sum_(n=0)^infinity (-1)**n (x**(2n+1))/(10**n) sum_(n=0)^infinity (-1)**n 10**n x**(2n) 10 sum_(n=0)^infinity (-1)**n x**(2n+1) sum_(n=0)^infinity (-1)**n 10**(n+1) x**(n+1) Correct: Your answer is correct. (b) Determine the interval of convergence.

Answer 1

One way to do this is to recall that for $|x|<1$, we have

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

so that

$\displaystyle\frac x{10x^2+1}=\frac x{1-(-10x^2)}=x\sum_{n=0}^\infty(-10x^2)^n=\sum_{n=0}^\infty(-10)^nx^{2n+1}$

(which seems to match the first option) so long as $|-10x^2|=10x^2<1$, or $-\frac1{\sqrt{10}}<x<\frac1{\sqrt{10}}$, which is the interval of convergence.