 
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 evendegree coefficients are zero.
2. Observe that /4 = arctan 1; in other words, = 4 arctan1. Use the
highestdegree 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
