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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

  

Source: Achar, Pramod - Department of Mathematics, Louisiana State University

 

Collections: Mathematics