AP Calculus BC · Infinite sequences and series

Start from the geometric series for 1/(1+u)

Problem

Start from the geometric series for $1/(1+u)$. (a) Substitute $u=x^{2}$ to obtain the Maclaurin series for $1/(1+x^{2})$. State its radius of convergence. (b) Integrate term by term from $0$ to $x$ to obtain the Maclaurin series for $\arctan x$. Test both endpoints $x=\pm 1$ and identify the two endpoint sums. (c) Approximate $\arctan(1/2)$ by the first four nonzero terms of the series. Using the alternating-series estimation theorem, bound the absolute error and determine the sign of the error. (d) A student argues that because $\arctan x$ is defined and smooth at $x=2$, the Maclaurin series must converge at $x=2$. Audit the argument by testing the series at $x=2$.

Hint

Geometric in $-x^{2}$, then integrate, is the Maclaurin series for arctangent.

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