Thread: write a function as a power series

1. write a function as a power series

f(x)=x/(9 + x^2)

We know that 1/(1-x) = (sigma)(0 to infinity)x^n

find f(x) and the interval of convergence.

2. Originally Posted by sprinks13
f(x)=x/(9 + x^2)

We know that 1/(1-x) = (sigma)(0 to infinity)x^n

find f(x) and the interval of convergence.
I will give you a hint this is $\displaystyle f(x)=x\cdot\int\frac{1}{3^2+x^2}$

and to find the interval of convergence just take $\displaystyle \lim_{n \to {\infty}}\frac{a_{n+1}}{a_n}<1$ and solve for x

3. I have managed to solve it:

$\displaystyle {a_n}=\frac{(-1)^nx^{2n+1}}{9^{n+1}}$

But I didn't use your hint. Now I am curious, how did you go about solving this?

4. Hello,

The most direct way is to say that :

$\displaystyle \frac{x}{9+x^2}=\sum_{n \geq 0} a_n x^n$

And to solve for $\displaystyle a_n$

What mathstud meant is to observe... You know the power series for $\displaystyle \arctan x$, whose derivative is $\displaystyle \frac{1}{1+x^2}$