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{√(10)}<x<\frac1{√(10)}`, which is the interval of convergence.