## 8.4. Taylor Series

### Proposition 8.4.16: Taylor Series for the Arc Tan

Note that the series converges at both endpoints.arctan(x) = x - 1/3 x^{3}+ 1/5 x^{5}- 1/7 x^{7}+ ... = for |x| 1

Just as with the series representation of *ln(1+x)* we would
start with a geometric series and integrate both sides. The details are
left to you .... the boundary points *|x| = 1* are an
application of Abel's Limit theorem, which you should be able to do as well.

f(x) = arctan(x)

### Fun Facts:

*/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ...*

Since *tan(/4) = 1* we have
*arctan(1)=/4*. Thus, this
statement follows from the above representation for *x = 1*.

Series approaching Pi/4=0.785398