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Chapter 1: Crises in Mathematics: Fourier's Series

Chapter 2: Infinite Summations

2.1 Avoiding Infinite Series

2.2 The Geometric Series

2.3 Calculating Pi

2.4 The Harmonic Series

2.5 Taylor Series

Taylor's Formula

d'Alembert

>Lagrange's Remainder

2.6 Emerging Doubts

2.5 Taylor Series (continued)

Lagrange's Remainder

Joseph Louis Lagrange was born in Turin, Italy under the name Giuseppe Lodovico Lagrangia. When he published his first mathematics in 1754, he signed it Luigi De la Grange Tournier (the final name being a reference to his native city). Shortly thereafter, he adopted the French form of his name, Joseph Louis. Like many people of his time, he was not consistent in his signature. “J. L. de la Grange” was the most common. It was only after his death that “Joseph Louis Lagrange” became the common spelling. His mathematical reputation was established at an early age, and he was a frequent correspondent of Euler and d'Alembert. In 1766 he succeeded Euler at the Berlin Academy, and in 1787 he moved to Paris. He was clearly the dominant member of the committee that met, probably in early 1808, to reject Fourier's treatise on the propagation of heat.

In the revised edition of Théorie des fonctions analytiques, published in the year of his death, Lagrange gives a means of estimating the size of the error that is introduced when any partial sum of a Taylor series is used to approximate the value of the original function. In other words, he finds a way of bounding the difference

What he shows is that this difference is

where c is some number (usually unknown) between a and x. While we do not know the value of c, the fact that it lies between a and x may be enough to find limits for the size of the error.

Lagrange's remainder enables us to answer three questions left open by d'Alembert's analysis of the binomial series:

1. What happens when x = 1?
2. If the series converges, how many terms must we take in order to obtain the desired degree of accuracy?
3. If the series diverges, how accurate can we be?

To simplify our calculations, we shall restrict our attention to the power series expansion of that d'Alembert studied:

If we take the partial sum up to

then the difference between this partial sum and is

When x is positive, we can find an upper bound on the absolute value of by taking c=0. The error that is introduced by using the polynomial approximation of degree n-1 is bounded by

Using Wallis's inequality, equation (2.3.4.1), we have that

and so

(2.5.4)

When x=1, the error term does approach zero as n gets larger. Given |x| < 1 and a limit on the size of the allowable error, inequality (2.5.4) can be used to see how large n must be. If |x| is larger than 1, then the error will eventually grow without bound. This bound is minimized when

If x = 200/199, we want to choose n = 300. The resulting approximation will be within

The Lagrange remainder for the Taylor series is more than a tool for estimating errors. It makes precise the difference between the polynomial and the target function that the polynomial approximates. This precision will come to play a critical role as we try to pin down the reasons why certain series behave well while others must be treated with great care.

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