1. ## [SOLVED] Maclaurin Series

I was trying to do this problem in my book

find the Maclaurin Series for atan(x^2) using the table found on the previous page

The table listed the series for atan(x) and they apparently wanted you to plug x^2 in for x to use the table to create the series. I am trying to figure out why you can make this substitution - I know that it works but the reason why didn't jump out at me. So I tried to use the definition of Maclaurin series,

$
\sum\limits_{n = 0}^\infty {\frac{{f^{(n)} (0)}}
{{n!}}} x^n = f(x)
$

to figure out why it was ok, but I couldn't. So my question is, why can you substitute another function of x into the definition of a Maclaurin and/or Taylor Series?

Here is what I tried to do - I let u(x) be x^2 and tried to show the following

$
\sum\limits_{n = 0}^\infty {\frac{{f^{(n)} (u(0))}}
{{n!}}} x^n = \sum\limits_{n = 0}^\infty {\frac{{f^{(n)} (0)}}
{{n!}}} u^n
$

but I don't know enough about these series to work that out

$\frac{1}{1-x} = 1 + x + x^2 + x^3 + ...$

$\frac{1}{1 + x^2} = \frac{1}{1 - (-x^2)} = 1 - x^2 + x^4 - x^6 + ...$

integrate ...

$\arctan(x) = C + x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + ...$

since $\arctan(0) = 0$ ... $C = 0$

so ...

$\arctan(x^2) = x^2 - \frac{x^6}{3} + \frac{x^{10}}{5} - \frac{x^{14}}{7} + ...$

3. Ty, but my question was why can you do the last step - I solved the problem I just don't know why you can take atan(x) and put in x^2 for x.

4. Originally Posted by billa
Ty, but my question was why can you do the last step - I solved the problem I just don't know why you can take atan(x) and put in x^2 for x.
because it's a composite function.

5. Wow, what was going on with my brain? I don't even know what I was trying to do there - good thing I didn't ask my teacher lol