Example 6.5.10
Find a power series representation for
![$ f(x) = \tan^{-1}(x)$](img1204.png)
.
Notice that
which has radius of convergence
![$ R=1$](img1206.png)
, since the above series
is valid when
![$ \vert-x^2\vert<1$](img1207.png)
, i.e.,
![$ \vert x\vert < 1$](img1168.png)
.
Next integrating, we find that
for some constant
![$ c$](img72.png)
.
To find the constant, compute
![$ c = f(0) = \tan^{-1}(0) = 0$](img1209.png)
.
We conclude that
Example 6.5.11
We will see later that the function
![$ f(x) = e^{-x^2}$](img1211.png)
has power series
Hence
This despite the fact that the antiderivative of
![$ e^{-x^2}$](img1214.png)
is not an
elementary function (see Example
![[*]](/usr/share/latex2html/icons/crossref.png)
).