The Maclaurin series expansion for tan1x where 1 lessthan x

The Maclaurin series expansion for tan^-1(x) where -1 lessthan x lessthan 1 is: tan^1(x) = x - x^3/3 + x^5/5 - x^7/7 +... Please use it to compute an estimate of ran^-1 (0.7) by hand (with aid of a hand-held calculator). Please include step-by-step calculation for the approximations by adding one term at a time (up to 4 terms) and calculate the absolute value of the approximate percent relative error (epsilon_a) at each step when appropriate. Please show your work for the three epsilon_a values calculated. Please develop a MATLAB M-file function program named atanx based on the above Maclaurin series to carry out computation for an estimate of tan^-1 (0.7) with the epsilon_a meeting a stopping criterion (epsilon_s) conforming to two significant figures. Please list your M-file program, the computed approximate percent relative error epsi _ alpha values at each step and the computed final estimated approximation to tan^-1 (0.7) should be printed out by your program. For example, using the function for an estimate of tan^-1(0.7) with two significant figures stopping criterion in command window can be: GG nfigure=2; GG atanx(0.7, nfigure)

Solution

(a) y tan^-1(x) = x - x³/3 + x⁵/5 - x⁷/7 - x⁹/9 ......

value of tan^-1(0.7) = 0.6107259644

step 1 y₁ = 0.7 error = - 0.0892740356

step 2 y₂ = 0.5856666667 error = 0.0250592977

step 3 y₃ = 0.619280667 error = - 0.0085547023

step 4 y₄ = 0.607515767 error = 0.003210177