[Math] Find a power series representation for the function and determine the interval of convergence

Question

Find a power series representation for the function $f\left( x \right)=\frac { x }{ 2{ x }^{ 2 }+1 } $ and determine of convergence.

I ended up with the following:

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

I don't know if that's right, and I also don't know what my interval of convergence would be.

$\left( -\frac { 1 }{ 2 } ,\frac { 1 }{ 2 } \right) $ perhaps?

Answer 1

Your answer is correct. One may recall that $$ \frac1{1+u}=\sum_{n=0}^\infty (-1)^n u^n, \quad |u|<1, $$ giving, for $2x^2<1$, $$ \frac1{1+2x^2}=\sum_{n=0}^\infty (-2)^n x^{2n} $$ that is

$$ \frac{x}{1+2x^2}=\sum_{n=0}^\infty (-2)^n x^{2n+1}, \quad x \in \left(-\frac{\sqrt{2}}2,\frac{\sqrt{2}}2\right). $$