18.014ESG Problem Set 10 Pramod N. Achar Summary: 18.014­ESG Problem Set 10 Pramod N. Achar Fall 1999 Wednesday 1. Compute the first few Taylor polynomials of arctanx about 0. You should find that all the even-degree coefficients are zero. 2. Observe that /4 = arctan 1; in other words, = 4 arctan1. Use the highest-degree Taylor polynomial you found in the preceding question to get an estimate of the value of ; then, use Taylor's Theorem to get a bound on the error in your estimate. Of course, unless you are much more arithmetically talented than I, you will use a calculator to help evaluate the Taylor polynomial. But back in the old days, before electronic computing equipment was omnipresent, people actually used Taylor polynomials to compute the value of . Using the above method, you would need a Taylor polynomial of huge degree in order to get an estimate that's correct to just a couple decimal places. (By my quick and possibly erroneous computations on a calculator, the 51st -degree Taylor polynomial gives a value of correct to one decimal place.) But in 1704, John Machin observed that Collections: Mathematics