sequences and series - Check that $f(x)=sum_{n=1}^{infty}x^2(1-x^2)^{n-1}$ is continuous or not.

Define $f:[0,1]\to \Bbb R$ by

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

Check that $f$ is continuous or not.

Attempt: $$f(x)=\sum_{n=1}^{\infty}x^2(1-x^2)^{n-1}\\
=\lim\int_{0}^{1}x^2(1-x^2)^{n-1}$$
Now, Putting $x=\sin\theta$, then $\mathrm dx=\cos\theta\mathrm d\theta$, therefore integral reduces to

$$\lim\int_{0}^{\pi/2}\sin^2\theta(1-\sin^2\theta)^{n-1}\cos\theta\mathrm d\theta\\
=\lim\int_{0}^{\pi/2}\sin^2\theta\cos^n\theta\mathrm d\theta$$

Now from here the result will depend on $n$ i.e. $n=2m,n=2m+1$
In these two cases the result will be different, Hence $f$ is not continuous.

am I right? Different approaches are invited. Thank you.