Geometric series


In mathematics, a geometric series is a series summing the terms of an infinite geometric sequence, in which the ratio of consecutive terms is constant. For example, the series is a geometric series with common ratio, which converges to the sum of. Each term in a geometric series is the geometric mean of the term before it and the term after it, in the same way that each term of an arithmetic series is the arithmetic mean of its neighbors.
While Greek philosopher Zeno's paradoxes about time and motion have been interpreted as involving geometric series, such series were formally studied and applied a century or two later by Greek mathematicians, for example used by Archimedes to calculate the area inside a parabola. Today, geometric series are used in mathematical finance, calculating areas of fractals, and various computer science topics.
Though geometric series most commonly involve real or complex numbers, there are also important results and applications for matrix-valued geometric series, function-valued geometric series, adic number geometric series, and most generally geometric series of elements of abstract algebraic fields, rings, and semirings.

Definition and examples

The geometric series is an infinite series derived from a special type of sequence called a geometric progression. This means that it is the sum of infinitely many terms of geometric progression: starting from the initial term, and the next one being the initial term multiplied by a constant number known as the common ratio. By multiplying each term with a common ratio continuously, the geometric series can be defined mathematically as
The sum of a finite initial segment of an infinite geometric series is called a finite geometric series, expressed as
When it is often called a growth rate or rate of expansion. When it is often called a decay rate or shrink rate, where the idea that it is a "rate" comes from interpreting as a sort of discrete time variable. When an application area has specialized vocabulary for specific types of growth, expansion, shrinkage, and decay, that vocabulary will also often be used to name parameters of geometric series. In economics, for instance, rates of increase and decrease of price levels are called inflation rates and deflation rates, while rates of increase in values of investments include rates of return and interest rates.
When summing infinitely many terms, the geometric series can either be convergent or divergent. Convergence means there is a value after summing infinitely many terms, whereas divergence means no value after summing. The convergence of a geometric series can be described depending on the value of a common ratio, see. Grandi's series is an example of a divergent series that can be expressed as, where the initial term is and the common ratio is ; this is because it has three different values.
Decimal numbers that have repeated patterns that continue forever can be interpreted as geometric series and thereby converted to expressions of the ratio of two integers. For example, the repeated decimal fraction can be written as the geometric series where the initial term is and the common ratio is.

Convergence of the series and its proof

The convergence of the infinite sequence of partial sums of the infinite geometric series depends on the magnitude of the common ratio alone:
  • If, the terms of the series approach zero and the sequence of partial sums converge to a limit value of.
  • If, the terms of the series become larger and larger in magnitude and the partial sums of the terms also get larger and larger in magnitude, so the series diverges.
  • If, the terms of the series become no larger or smaller in magnitude and the sequence of partial sums of the series does not converge. When, all the terms of the series are the same and the grow to infinity. When, the terms take two values and alternately, and therefore the sequence of partial sums of the terms oscillates between the two values and 0. One example can be found in Grandi's series. When the common ratio is the imaginary unit and, the partial sums circulate periodically among the complex numbers,,,,,,,,..., never converging to a limit. When the common ratio is a root of unity for a rational number in lowest terms and with any, the partial sums of the series will circulate indefinitely with a period of, never converging to a limit.
The rate of convergence shows how the sequence quickly approaches its limit. In the case of the geometric series—the relevant sequence is and its limit is —the rate and order are found via
where represents the order of convergence. Using and choosing the order of convergence gives:
When the series converges, the rate of convergence gets slower as approaches. The pattern of convergence also depends on the sign or complex argument of the common ratio. If and then terms all share the same sign and the partial sums of the terms approach their eventual limit monotonically. If and, adjacent terms in the geometric series alternate between positive and negative, and the partial sums of the terms oscillate above and below their eventual limit. For complex and the converge in a spiraling pattern.
The convergence is proved as follows. The partial sum of the first terms of a geometric series, up to and including the term,
is given by the closed form
where is the common ratio. The case is merely a simple addition, a case of an arithmetic series. The formula for the partial sums with can be derived as follows:
for. As approaches 1, polynomial division or L'Hôpital's rule recovers the case.
As approaches infinity, the absolute value of must be less than one for this sequence of partial sums to converge to a limit. When it does, the series converges absolutely. The infinite series then becomes
for.
This convergence result is widely applied to prove the convergence of other series as well, whenever those series's terms can be bounded from above by a suitable geometric series; that proof strategy is the basis for the ratio test and root test for the convergence of infinite series.

Connection to the power series

Like the geometric series, a power series has one parameter for a common variable raised to successive powers corresponding to the geometric series's , but it has additional parameters  one for each term in the series, for the distinct coefficients of each , rather than just a single additional parameter  for all terms, the common coefficient of  in each term of a geometric series. The geometric series can therefore be considered a class of power series in which the sequence of coefficients satisfies  for all  and .
This special class of power series plays an important role in mathematics, for instance for the study of ordinary generating functions in combinatorics and the summation of divergent series in analysis. Many other power series can be written as transformations and combinations of geometric series, making the geometric series formula a convenient tool for calculating formulas for those power series as well.
As a power series, the geometric series has a radius of convergence of 1. This could be seen as a consequence of the Cauchy–Hadamard theorem and the fact that for any or as a consequence of the ratio test for the convergence of infinite series, with implying convergence only for However, both the ratio test and the Cauchy–Hadamard theorem are proven using the geometric series formula as a logically prior result, so such reasoning would be subtly circular.

Background

2,500 years ago, Greek mathematicians believed that an infinitely long list of positive numbers must sum to infinity. Therefore, Zeno of Elea created a paradox, demonstrating as follows: in order to walk from one place to another, one must first walk half the distance there, and then half of the remaining distance, and half of that remaining distance, and so on, covering infinitely many intervals before arriving. In doing so, he partitioned a fixed distance into an infinitely long list of halved remaining distances, each with a length greater than zero. Zeno's paradox revealed to the Greeks that their assumption about an infinitely long list of positive numbers needing to add up to infinity was incorrect.
Euclid's Elements has the distinction of being the world's oldest continuously used mathematical textbook, and it includes a demonstration of the sum of finite geometric series in Book IX, Proposition 35, illustrated in an adjacent figure.
Archimedes in his The Quadrature of the Parabola used the sum of a geometric series to compute the area enclosed by a parabola and a straight line. Archimedes' theorem states that the total area under the parabola is of the area of the blue triangle. His method was to dissect the area into infinite triangles as shown in the adjacent figure. He determined that each green triangle has the area of the blue triangle, each yellow triangle has the area of a green triangle, and so forth. Assuming that the blue triangle has area 1, then, the total area is the sum of the infinite series
Here the first term represents the area of the blue triangle, the second term is the area of the two green triangles, the third term is the area of the four yellow triangles, and so on. Simplifying the fractions gives
a geometric series with common ratio and its sum is:
In addition to his elegantly simple proof of the divergence of the harmonic series, Nicole Oresme proved that the arithmetico-geometric series known as Gabriel's Staircase,
In the diagram for his geometric proof, similar to the adjacent diagram, shows a two-dimensional geometric series. The first dimension is horizontal, in the bottom row, representing the geometric series with initial value and common ratio
The second dimension is vertical, where the bottom row is a new initial term and each subsequent row above it shrinks according to the same common ratio, making another geometric series with sum,
This approach generalizes usefully to higher dimensions, and that generalization is described above in.