An infinite sequence is an ordered listing of an infinite number of real numbers, such as ½, ¼, 1/8, 1/16, ... In more formal mathematical language, an infinite sequence is a function whose domain is the set of natural numbers (sometimes with 0 included) and whose range is the set of real numbers. An infinite series is just the sum of all the terms of an infinite sequence. For example, 1/2+1/4+1/8+...+1/2n+... designates the infinite series whose nth term has the form 1/2n. In general if the nth term of an infinite series is an, then the series is written a1+a2+a3+...+an+..., which is also written as ∑an where the capital Greek letter ∑ stands for the summation of all terms of the form an. Of interest to mathematicians is which infinite series have finite sums and which grow without bound as each new term is added. It may surprise someone who is unfamiliar with infinite series to hear that some of them actually do have finite sums. A naïve view might be that because they have an infinite number of terms being added together their sums must always be infinite; but consider the infinite series above: 1/2+1/4+1/8+...+1/2n+.... Mathematicians define a sequence of "partial sums", S1, S2, S3, ... where, in general, Sn is the sum of the first n terms of the infinite series. Thus, for our series, S1=1/2, S2=3/4, S3=7/8, and, in general, Sn=(2n-1)/2n=1-1/2n. Now as n grows without bound, Sn gets ever closer to 1. Clearly Sn can never become greater than 1, so mathematicians agree that the sequence of partial sums for this infinite series approaches 1 and this is defined to be the sum of the series 1/2+1/4+1/8+...+1/2n+.... Thus, in general, the sum of an infinite series is the limit of the sequence of partial sums as n grows without bound, provided that such a limit exists. In this case, the infinite series is said to converge to this limit. If the sequence of partial sums continues to grow without bound as n grows without bound, then the infinite series does not have a sum and is said to diverge. In calculus, a good deal of time is devoted to determining which infinite series converge and which do not. The problem is that it is not always simple or even possible to get a formula for the sequence of partial sums, so mathematicians have needed to devise a number of convergence tests and to prove that they work for certain classes of infinite series.
The reason that questions of convergence are so important is that infinite series are used as representations of many of the most familiar mathematical functions to approximate values of these functions. Such approximations are valid only if they are carried out within the interval of convergence of the series being used.
The interval of convergence is just the set of all real numbers for which the infinite series converges. The use of infinite series as function representations is due to the work of the English mathematician Brook Taylor (1685-1731), who showed how to construct such series using techniques of calculus. Such series are now called Taylor series. An example is the Taylor series for the familiar sine function from trigonometry. Using Taylor's methods, one can show that the infinite series representation of sin(x) is x-x3/3!+x5/5!-x7/7!+...+(-1)n+1x2n+1/(2n+1)!+...,where n=0,1,2,3.... It can also be shown that the interval of convergence for this infinite series is the set of all real numbers. Algorithms based upon series expansions of functions are used in calculators and computers to give highly accurate approximations of those functions. Thus when one pushes the sin button on a calculator followed by a number, the calculator does not search in some infinite table of values to find the sine of just this number. Rather it computes the sum of a sufficient number of terms of a series expansion for the sine function. Taylor's theory also allows the mathematician or engineer to determine how many terms of an infinite series expansion must be used to get a desired number of decimal places of accuracy.
Infinite series may be used to derive a number of remarkable mathematical results. For example, it can be shown that the Taylor series expansion of the inverse tangent function, denoted arctan(x) is x-x3/3+x5/5-x7/7+... Now the inverse tangent of 1 is /4, which means that /4=1-1/3+1/5-1/7+...or =4-4/3+4/5-4/7+... Early attempts to give to a specifed number of decimal places made use of this infinite series. The only problem with this method is that this infinite series converges to very slowly, so that modern computer programs for generating millions of decimal places of use more sophisticated algorithms. On a different topic, a famous paradox posed by the Greek philosopher Zeno (5th century BC) can be explained by the theory of infinite series. Zeno said that if one is standing at point A one meter from point B, one can never reach point B by traversing the line segment connecting points A and B. Zeno's reasoning was that to get from A to B, one would first have to travel half the distance or ½ meter, then half the remaining distance or ¼ meter, then half again or 1/8 meter, and again and again and again..., so that one's total distance would be computed by ½+1/4+1/8+1/16+... Since this is the sum of an infinite number of terms, Zeno claimed that the distance could never be traversed. As we saw above, however, the partial sums of Zeno's infinite series approach 1 as a limit, so that the by the mathematical definition of the sum of an infinite series, this total distance is 1. The subject of infinity has vexed mathematicians throughout recorded history, prompting them to seek carefully thought out explanations for the paradoxes, such as Zeno's, that occur when the term "infinity" is not treated with precise mathematical definitions. The work of Taylor and other mathematicians in infinite series has been just one example of this quest to understand the infinite.
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