Sequences and series

Summary: Sequences and series are important mathematical representations with numerous, interesting applications.

A sequence is a list of objects, called “terms,” arranged in a fixed pattern such as 1, 3, 5, 7, 9,… or Monday, Tuesday, Wednesday, Thursday, Friday,…. In a series, the terms of a sequence are typically added together. Series have a long history of being used to approximate functions or represent geometric quantities. For example, in the seventeenth century, James Gregory showed how the areas of a circle and hyperbola could be obtained using series. In the early days of calculus, series represented geometric quantities and were manipulated using methods extended from finite procedures. Mathematicians like Niels Abel critiqued the rigor of series and expressed concerns with the foundations of calculus. The theory of series was later made rigorous within the field of analysis. Series are important to many areas in science and engineering. Sequences are explored in the primary and middle grades, while series are introduced in high school.

Famous Sequences

One very famous sequence emerges when considering the reproductive habits of rabbits. Consider two rabbits that are too young to reproduce after their first month of life but can and do reproduce after their second month of life. That pair of rabbits produces another pair after its second month and for each month thereafter. If one assumes that none of the rabbits die and that each pair reproduces in the same manner as the first, the number of pairs of rabbits at the end of each month corresponds to the elements of the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34,… . This sequence is known as the Fibonacci number. It is named after Leonardo de Pisa who was called Fibonacci, a nickname meaning “son (filius) of Bonaccio.” He wrote about it in his 1202 book Liber Abaci. With the exception of the first two terms, each successive term is found by adding the two terms prior to it. This sequence appears in nature in other situations, including the arrangement of leaves on the stems of certain plants, the fruitlets of a pineapple and the spirals of shells. Some mathematical historians suggest that a Fibonacci-like sequence of integers is also represented in stone balance weights excavated in the 1960s that originated in the eastern Mediterranean during the Late Bronze Age.

Other specific types of sequences have been explored. In 1940, Pavel Aleksandrov introduced a concept called “exact sequences,” which found relevance in a wide variety of mathematical fields. In 1954, Jean-Pierre Serre was awarded a Fields Medal, the most prestigious award in mathematics, in part because of his work on spectral sequences.

Series

A series is often the sum of the terms of a sequence. Series originate as early as the Indian mathematician and astronomer Brahmagupta who gave rules for summing series in his 628 c.e. work Brahmasphutasiddanta (The Opening of the Universe). The sum of the terms of an arithmetic sequence is called an arithmetic series. The arithmetic series 1+2+3+4+. . . +97+98+99+100 is also a well known one, as it is related to mathematician Carl Friedrich Gauss (1777–1855). At a very young age (around 6 years old), Gauss found the sum of the natural numbers (1, 2, 3, 4, …) from 1 to 100. That is, the sum given by the series 1+2+3+4+. . . +97+98+99+100. He was given this task by his teacher to keep him busy while the teacher worked with the other students in the class who were not as mathematically gifted as Gauss. After a relatively short time, Gauss returned to the teacher with the sum 5050. Gauss’s method was to pair up the terms of the series. Taking the sum of the first and last term (1+100) yields 101. This is the same as the sum of the second and second to last (2+99=101), the third and third to last (3+98+101), and so forth. In all, there are 50 such pairs, each of which sums to 101. Thus,

1 + 2 + 3 + 4 + … + 97 + 98 + 99 + 100

 =(50)(101) = 5050.

Many mathematicians advanced the theory of important series such as power series, trigonometric series, Fourier series, and time series. For example, Nicholas Mercator represented the function log(1+x) as a series in 1651. Taylor series, named after Brook Taylor, is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. From the early history of analysis, these power series were important in the study of transcendental functions. Data given as a sequence of data points over time led Wilhelm Lexis to develop time series in 1879.

Applications of Series

Some other series arose in the context of questions related to physics and sparked controversy. The mathematics and physics of a vibrating string and solutions of the wave equation led to trigonometric series. Daniell Bernoulli, Jean Le Rond d’Alembert, Leonhard Euler, and Joseph-Louis Lagrange debated the nature of trigonometric series in the eighteenth century. Joseph Fourier developed Fourier series for the heat equation in the nineteenth century, which was criticized at the time because it contradicted a theorem by Augustin-Louis Cauchy but was explored more rigorously by Johann Dirichlet. An overshoot or ringing in Fourier series was first observed by H. Wilbraham and later explored by Josiah Gibbs. The Gibbs phenomenon has implications in signal processing. The three-body problem, which investigates the behavior and stability of three mutually attracting orbiting bodies in the solar system, was solved by Delaunay in 1860 via representing the longitude, latitude, and parallax of the moon as an infinite series.

However, in 1892, Jules Henri Poincaré showed that these and similar solutions were not in general uniformly convergent, and this criticism created doubt about proofs of the stability of the solar system and eventually led to the formation of the field of deterministic chaos. A prize was offered by King Oscar II of Sweden for a solution to the extension of the three-body problems to n bodies. It has since been proven that no general solution is possible, but the n-body problem was also connected to series in Quidong Wang’s 1991 work.

Series were also important as mathematicians searched for efficient ways to represent π and find its digits. Keralese mathematician Madhava of Sangamagramam may have been the first when he used 21 terms of a series and stated π correctly to 11 places. In the 1800s, William Shanks used a series to calculate digits of π in the morning and check them in the evening. He calculated 707 digits of π using this method. However, there was a suspicious lack of the number “7” in the last digits, and it was later found that only the first 527 digits were correct. Johann Lambert used the same series to show in 1761 that π must be irrational—it cannot be expressed as a ratio of whole numbers and has an infinite, non-repeating decimal expansion. Srinivasa Ramanujan found series that converged more rapidly than others, and these efficient series were used as the foundations of computer algorithms.

Binary Series

A very famous series is the binary series that consists of powers of 2: 2⁰+2¹+2²+2³+2⁴+2⁵+… . It is theorized that the King of Persia, finding himself very bored, asked that a game be invented for his amusement. The inventor of the game the king found most enjoyable would be given a reward. A servant of the king created the game of chess that was most pleasing to the king. When asked what prize he would like, the servant replied that he wanted grains of rice. The chessboard consists of 64 small squares. As a reward the servant asked for 1 grain of rice for the first square, 2 for the second square, 4 for the third square, 8 for the fourth square and so forth, until all 64 squares had been accounted for. The number of grains of rice requested is the sum 2⁰+2¹+2²+2³+2⁴+2⁵+… 2⁶³, and it amounts to 274,877,906,944 tons of rice, which is more rice than has been cultivated on Earth since recorded time. The story goes that the king grew furious at the servant once he knew what was requested. The servant was taking the rice as he received it and distributing it among the poor. At some point, the king indicated that he did not have the rice to pay the servant. The servant indicated that he was content with the amount that he had already received and that it was the king who offered a reward not he who made the initial request. Both parties were pleased.

Applications in Economics

Series appear in many other contexts as well. For example, the future value of an ordinary annuity can be found using a series. An ordinary annuity is an account where an individual makes identical deposits on a regular schedule. The money in the account earns interest that is compounded with the same frequency as the deposits. Suppose an individual deposits $100 every year into an account that earns 6% interest annually. Three years later, the first year’s deposit has earned interest over two years, the second account over one year, and the last deposit not at all. The money in the account after three years is given by: 100(1.06)⁰+100(1.06)¹+100(1.06)². The general series can be expressed as a single number

where A is the future value of the annuity, P is the payment made at the end of each period, i is the interest rate per period, and n is the number of periods.

Limits

Though infinite sequences consist of infinitely many terms, it may be the case that the sum of the terms of such sequences converges on a given value. Such is the case of the geometric series .9+.09+.009+.0009+… .In this series, the first term is .9 and the common ratio is

Applying the formula for the sum of the first n terms of the series yield

As the number of terms approaches infinity (as n→∞), the fraction

becomes so small that one may consider it zero. Therefore,

as n grows infinitely large. Since

S_n = .9 + .09 + .009 + .0009 + … = .9∞

one arrives at the very famous result that .9∞=1.

Bibliography

Ferraro, Giovanni, and Marco Panza. “Developing Into Series and Returning From Series: A Note on the Foundations of Eighteenth-Century Analysis.” Historia Mathematica 30, no. 1 (2003).

Klein, Judy. Statistical Visions in Time: A History of Time Series Analysis, 1662–1938. Cambridge, England: Cambridge University Press, 1997.

Kline, Morris. Mathematical Thought From Ancient to Modern Times. New York: Oxford University Press, 1972.

Laugwitz, Detlef. Bernhard Riemann, 1826–1866: Turning Points in the Conception of Mathematics. Boston: Birkhäuser Boston, 2008.