Given the Maclaurin series for arctan(x)= $$\sum_{k=0}^{\infty} (-1)^k * \frac{x^{2k+1}}{2k+1}$$
a. Find the interval of convergence of this power series.
b. Discuss the meaning of this interval for the series as compared to the graph of y=arctan(x).
I found the interval of convergence through the ratio test: it is [-1,1].
Not sure about the second part. Can someone explain to me what it actually does?
Look at the McLauren's series of arctan(x).
$$ arctan(x) = \int _0^x \frac {1}{1+t^2} dt =$$
$$ \int _0^x (1-t^2+t^4-t^6+...)dt = $$
$$ x-x^3/3 +x^5/5 - x^7/7 + ....$$
Compare it with your series.
Note that the graph of arctan(x) is bounded over the entire real line but your series does not converge beyond its interval of convergence.
Thus the equality of the function and its MacLauren's series is conditional.