The inverse-tangent function can be expressed for as x = _ n=0 ^ (-1)^ n x^ 2n+1 2n+1
MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
The inverse-tangent function can be expressed for ∣x∣≤1 as
arctanx=n=0∑∞(−1)n2n+1x2n+1.
A student wishes to approximate arctan(0.3) using a finite number of terms from this series.
(a) Write explicitly the first four non-zero terms of the series representation for arctan(0.3). Do not simplify the numerical values.
(b) Determine the least integer N such that evaluating the partial sum SN guarantees an approximation to arctan(0.3) whose error has magnitude less than 10−7. Provide a complete justification, including any inequalities used.
(c) Compute SN from part (b) and record the resulting decimal approximation of arctan(0.3) rounded to eight decimal places. State the maximum possible error associated with this approximation.