3. Selected Problems from the History of the Infinite Series

3.1 Introduction

Mathematicians have been intrigued by Infinite Series ever since antiquity. The question of how an infinite sum of positive terms can yield a finite result was viewed both as a deep philosophical challenge and an important gap in the understanding of infinity. Infinite Series were used throughout the development of the calculus and it is thus difficult to trace their exact historical path. However, there were several problems that involved infinite series that were of significant historical importance. This section contains selected problems that represent an introduction to the historical significance of the Infinite Series.

3.2 James Gregory's Infinite Series for arctan

Most of Gregory’s work was expressed geometrically, and was difficult to follow. He had all the fundamental elements needed to develop calculus by the end of 1668, but lacked a rigorous formulation of his ideas. The discovery of the infinite series for arctan x is attributed to James Gregory, though he also discovered the series for tan x and sec x.
Here is how one can find the derivative of arctan x: The above is a modern proof, Gregory used the derivative of arctan from the work of others. The infinite series for can be found by using long division. Integrating this infinite series term-by-term produces, which is the infinite series for arctan.

Prior to Leibniz and Newton’s formulation of the formal methods of the calculus, Gregory already had a solid understanding of the differential and integral, which is shown here. Although the solution above is in modern notation, Gregory was able to solve this problem with his own methods. Gregory was one of the first to relate trigonometric functions to their infinite series using calculus, although he is primarily only remembered noted for finding the infinite series for the inverse tangent. (Boyer 429)

3.3 Leibniz's Early Infinite Series

One of Leibniz's earlier experiences with infinite series was to find the sum of the reciprocals of the triangular numbers, or .

By using partial fraction decomposition, the fraction can be split so that .

The first n terms of the series are: By factoring out the 2 and by rearranging the terms: all but the first and last term cancel, and the sum reduces to since .

Therefore, the sum of the reciprocals of the triangular numbers is 2.

This problem was historically significant as it served as in inspiration for Leibniz to explore many more infinite series. Since he successfully solved this problem, he concluded that a sum could be found of almost any infinite series. (Boyer, 446-447)

3.4 Leibniz and the Infinite Series for Trigonometric Functions

After having already developed methods for differentiation and integration, Leibniz was able to find an infinite series for sin(z) and cos(z). He began the process by starting with the equation for a unit circle: and differentiating with respect to x: By the equation of the unit circle given above and so Prior to Leibniz attempting to solve this problem, Newton had discovered the binomial theorem. Therefore, by simple application of Newton’s rule, Leibniz was able to expand the equation into an infinite series: Leibniz then integrated both sides. The right side of the equation can be integrated term-by-term and the left side of the equation is equal to arcsin(x). This can easily be shown: Therefore, integrating both sides yields: At this point, Leibniz had found the infinite series for arcsin(x), a result which Newton had found as well. Leibniz then used a process he and Newton both discovered independently: Series Reversion. That is, given the infinite series for a function, he found a way to calculate the infinite series for the inverse function.

In this case, the process worked by first taking the sin (the inverse function for arcsin) of both sides of the equation: Now Leibniz assumed that an infinite series for sin(y) exists that is of the form: Leibniz had said that sin y = x, therefore for each instance of x in he substituted the assumed infinite series for sin y. He knew that the result of substituting in the this series for x must yield y, as it was stated: Therefore, he knew the coefficient of the first term a1=1 and all of the other coefficients must add up to 0. In order to further explain, the first 3 coefficients of the expansion will be solved for. When the series is substituted, the only possible way to have a term is when it is substituted for x. The first term in the expansion will therefore be . There will be no term, and the term will be obtained by both the 3rd power y term being plugged into x, and the 1st power y term being plugged into thus yielding where the sum of the coefficients must be 0 (because there is no term left over in the expansion). The same process yields the equation for the 5th order term which is . At this point the resulting expansion is: Now equations for each coefficient can be set up: Solving the equations using the previous results in each calculation yields: Substituting the coefficients back into the assumed infinite series for sin y, he determined that: By simply differentiating this equation term-by-term Leibniz was also able to find the infinite series for cos y (Boyer and Merzbach 448 - 449).

Leibniz not only laid the groundwork for the Taylor series, but he (and simultaneously Newton) was the first to discover the series for these trigonometric functions. He invented his own method for finding the infinite series of a function’s inverse. For a more thorough description of the process of Series Reversion please refer to Eric's Treasure Trove of Mathematics on the Web.

3.5 Euler's Sum of the Reciprocals of the Squares of the Natural Numbers

Much work was done with infinite series by Euler. He was able to use infinite series to solve problems that other mathematicians were not able to solve by any methods. Neither Leibniz nor Jacques Bernoulli were able to find the sum of the inverse of the squares - they even admitted as much. The sum was unknown until Euler found it through the manipulation of an infinite series: In order to find this sum, Euler started by examining the infinite series for sin z. Equating sin z to zero gave Euler the roots of the infinite expansion .

That is, the roots of this equation are Now, left with the equation with roots dividing by z results in substituting yields and By using properties involving polynomials, it is known that the sum of the reciprocals of the roots is the negative of the coefficient of the linear term, assuming the constant term is 1. Applying this here, we get multiplying through by p 2, we get Which is the sum of the inverse of the squares. Starting with cos x instead of sin x, he obtained the sum for the sum of the squares of the odd natural numbers. He solved problems using infinite series that could not be done in any other way, and developed new ways to manipulate them. (Boyer 496-497)