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This programme discusses how the integral test is used to decide the convergence of series, and how partial sums of divergent series give a good way of approximating n factorial when n is large.
Metadata describing this Open University video programme
Module code and title: M203, Introduction to pure mathematics
Item code: M203; 02B
First transmission date: 1987
Published: 1987
Rights Statement:
Restrictions on use:
Duration: 00:24:38
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Producer: Pip Surgey
Contributors: David Brannan; Ian Harrison
Publisher: BBC Open University
Keyword(s): N Factorial; Partial Sums; Divergent series; Stirling's Formula; Wallis's formula
Subject terms: Estimation theory; Mathematical statistics; Probabilities
Footage description: What is the connection between calculating factorial n for large values of n and finding partial sums of the series log n? The answer is Stirling's Formula and that's what this programme sets out to prove, that n! is more or less equal to root 2 pi n times (n/e) to then n, certainly when n is large. The programme begins by examining the need for such an approximation, n! occurs in many probabilities, yet grows very rapidly and so is soon out of the range of most pocket calculators. Two problems are posed; the probabilities of dropping equal number of heads and tails using 200 coins and that of shuffling a pack of cards into a given order. The resulting expressions involve finding 100!, 200! and 52! Next the programme reviews the idea of integration as the (common) limit of sequences of upper and lower sums for various partitions. It establishes, by means of animation that monotonic functions are integrable, and that integrability is not dependent on continuity. This leads on to the Integral Test, If the partial sums of a series of positive terms is bounded, then the series converges. Associated with the series is a monotonic, decreasing function. The Integral Test asserts that the series converges if the integral of this function is bounded and diverges otherwise. The programme illustrates this by showing that the integral of 1 over x2 is bounded, so sigma l/n2 converges, but that the integral of 1 over x is unbounded, so sigma l/n does not converge. Clearly, by the same argument, sigma log n is not convergent, but it is possible to approximate partial sums of sigma log n by finding the averages of the upper and lower sums from 1 to n. It turns out that working out the error in taking this average by integrating log n leads to Stirling's Formula! The programme concludes by using Stirling's Formula to estimate the coins and card probabilities posed at the start of the problem. The programme should be of interest to anyone who has studied analysis (or calculus) to Advanced or first year undergraduate level; any viewer should be familiar with the ideas behind integration as well as the integrals of 1 over x2, lower x and log x, and with simple probability calculations.
Master spool number: HOU5694
Production number: FOUM271J
Videofinder number: 572
Available to public: no