Please show work. Thank you.
Taylor and MacClaurin Series Module Project In this Module Project we are going to approximate Ï to the limits of our calculators, and, in the process, see how Taylor and MacClaurin series make these kinds of approximations possible This project is broken down into sections, which, when completed, will result in a method of approximating the value of Ï that is only limited by the computational devise we have available to âcrunch the numbers. a) (10 points) First, we'll do some trigonometry show that 4 = tan-1-tan-1 3 by using the addition formula for tangent tan x + tany 1- tan x tan y tan(x + y) = (20 points) Use the MacClaurin Series for y = tan-1 x to state series whose sums are tan-1 , and tan11 respectively b) c) (20 points) Use parts a) and b) to express Ï in terms of these series d) (20 points) Use partial sums of these series to approximate Ï accurate to five decimal places. How many partial sums are needed to accomplish this approximation? e) (10 points) Approximate Ï accurate to the limits of your calculator, (or computer) f) (20 points) This worked because the fractionse the property that1. The series converges relatively quickly to Ï because these fractions are fairly close to 0, Do you think there are other pairs of fractions that would work like the two used here, and are even closer to zero? Would these fractions result in an even faster converging series for approximating Ï? Explain