Pi in the Sky

You are the "super-programmer" for the world's largest supercomputer. Your boss had taken you aside to give you your next assignment. He explains, confidentially, that for several days now, Project SETI -- which is concerned with the search for extraterrestrial intelligent life -- has been receiving a string of digits from a powerful point source near Tau Ceti. He suspects that these are the digits in the decimal expansion of . However, he doesn't know when they started transmitting, so they may have gotten quite far out, perhaps to the millionth decimal place or beyond. Your assignment is to calculate to more decimal places than ever before, in order to compare these computed digits with those from the extragalactic transmission. What you need is an algorithm; that is what you will develop here.

1. Express tan-1 z as the definite integral of some function f(t), with z as the upper limit. Show how, if you could obtain a numerical value for this integral when z=1, then you could get an exact numerical value for .

2. Now write the integrand f(t) as a geometric series. Use the identity for the sum of the first n terms of a geometric series to express the integrand as the sum of the first n terms plus an "error term."

3. Integrate this sum term by term to get an expression for tan-1z consisting of a polynomial plus an error term. Show that the error term has the form of an integral.

4. Give an upper bound on the error incurred when using only the polynomial of part (c) as your approximation to tan-1 z. How many terms do you need to give a value for accurate to a hundred decimal places? To a million? If the computation of each term and its addition to the previously computer terms takes 1 microsecond on the supercomputer, then how many years will it take to compute to a hundred decimal places? To a million? To understand why this is a poor series to compute , evaluate the first six polynomials at z=1.

5. You mention your problems over lunch to Sylvia, the mathematician down the hall. She jots a couple of formulas down for you:
 /4 = tan-1(1/2) + tan-1 (1/3) (1) /4 = 4 tan-1 (1/5) - tan-1 (1/239). (2)

Verify the correctness of these identities (It is not sufficient to punch the arctan button on your calculator! (do you see why not?))

1. How would you convert Sylvia's formulas into algorithms to approximate ? Do her formulas yield better approximations (in terms of time and money) than your original method in parts (a)-(d)? Evaluate the first six polynomials for both part of (1) and (2) and put these beside the six term approximation in part (d). Compare the number of terms of your original series and the series generated by formulae (1) and (2) needed to approximate to a million decimal places. How long will these formulae take on the supercomputer to yield approximations to correct to a million decimal places?

Slightly modified from: Student Research Projects in Calculus by Marcus Cohen, Edward D. Gaughan, Arthur Knoebel, Douglas S. Kurtz, and David Pengelley. The Mathematical Association of America, 1991.