I was given
$f(x) = x^2 \sqrt{1-x^2}$
I was first asked to get the chebyshev expansion between $[-1,1]$ i did that part.
Now i have to get the taylor expansion i am not sure how to do this because taylor expansion is at one point like
$x = x_a$
and then you can use the standard formula.
But now that i have been given a range $[-1,1]$ what would i do? i could potentially just use $-1$ but that would be only that point…
I am a bit confused.
Best Answer
Apply the Taylor expansion of $\sqrt{1-a}$ for $a<1$,
$$\sqrt{1-a}=1-\frac a2 - \frac {a^2}{8} - \frac {a^3}{16} ...$$
and let $a=x^2$,
$$\sqrt{1-x^2}=1-\frac {x^2}{2} - \frac {x^4}{8} - \frac {x^6}{16} ...$$
Then,
$$f(x) = x^2 \sqrt{1-x^2} = x^2-\frac {x^4}{2} - \frac {x^6}{8} - \frac {x^8}{16} ...$$