So as I started learning about Taylor series, I noticed that the $\arctan(x)$ expansion misses a factorial in the denominator. Since a Taylor series does have a factorial, I was wondering: how is the expansion of $\arctan(x)$ a Taylor expansion? Thanks in advance!
Also, how do I know how to choose alpha when calculating the expansion?
Best Answer
The Taylor expansion is not a very particular expansion, but a method to derive such an expansion. But suppose $f(x)=\sum_k a_k(x-x_0)^k$ and everything is nice enough (so that we might pull the differential under the series). Then
$$ f^{(k)}(x_0) = k!a_k $$
So if such a form exists it nescessarily has to satisfy $a_k = f^{(k)}(x_0)/k!$.
Thus the characteristic of a taylor expansion is not having a $k!$ but being in terms of $x-x_0$.
So why do you not see $k!$ in your form? Remember that $\arctan^{(k)}$ has something like $k!$ as a factor. So after cancelling this out you remain with just something polynomial.