# Math Help - power series representation

1. ## power series representation

What is the power series expression for f(x)=x^2sin(x). Find the radius of convergence too.

I know that the power series of sinx is (-1)^n (x^(2n+1))/(2n+1)! But what about the x^2? Do you just take the derivative of it?

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

then $\displaystyle x^2\sin{x} = x^2\sum_{n = 0}^{\infty}\frac{(-1)^nx^{2n+1}}{(2n+1)!} = \sum_{i=0}^{\infty}\frac{(-1)^nx^{2n + 3}}{(2n+1)!}$ (from the Distributive Law).

To check the radius of convergence use the ratio test.

3. If...

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

... multiplying both therms of (1) by $x^{2}$ You obtain...

$\displaystyle x^{2}\ \sin x= \sum_{n=0}^{\infty} (-1)^{n}\ \frac{x^{2n+3}}{(2n+1)!}$ (2)

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