is geometric, because each term except the first can be obtained by multiplying the previous term by .
Geometric series are one of the simplest examples of infinite series with finite sums. Historically, geometric series played an important role in the early development of calculus, and they continue to be central in the study of convergence of series. Geometric series are used throughout mathematics, and they have important applications in physics, engineering, biology, economics, computer science, queueing theory, and finance.
The terms of a geometric series form a geometric progression, meaning that the ratio of successive terms in the series is constant. The following table shows several geometric series with different common ratios:
|10||4 + 40 + 400 + 4000 + 40,000 + ···|
|1/3||9 + 3 + 1 + 1/3 + 1/9 + ···|
|1/10||7 + 0.7 + 0.07 + 0.007 + 0.0007 + ···|
|1||3 + 3 + 3 + 3 + 3 + ···|
|−1/2||1 − 1/2 + 1/4 − 1/8 + 1/16 − 1/32 + ···|
|–1||3 − 3 + 3 − 3 + 3 − ···|
The behavior of the terms depends on the common ratio r:
Consider the sum of the following geometric series:
This series has common ratio 2/3. If we multiply through by this common ratio, then the initial 1 becomes a 2/3, the 2/3 becomes a 4/9, and so on:
This new series is the same as the original, except that the first term is missing. Subtracting the new series from the original series cancels every term in the original but the first:
A similar technique can be used to evaluate any self-similar expression.
For , the sum of the first n terms of a geometric series is:
where a is the first term of the series, and r is the common ratio. We can derive this formula as follows:
The formula follows by multiplying through by a.
As n goes to infinity, the absolute value of r must be less than one for the series to converge. The sum then becomes
When a = 1, this simplifies to:
the left-hand side being a geometric series with common ratio r. We can derive this formula:
The general formula follows if we multiply through by a.
This reasoning is also valid, with the same restrictions, for the complex case.
And here is a geometric way of looking at this formula from E.Hairer and G.Wanner, Analysis by Its History, section III.2, FIGURE 2.1, page 188, Springer 1996:
Since (1 + r + r2 + ... + rn)(1 - r) = 1-rn+1 and rn+1 → 0 for | r | < 1, the limit is 1 /(1 − r).
A repeating decimal can be thought of as a geometric series whose common ratio is a power of 1/10. For example:
The formula for the sum of a geometric series can be used to convert the decimal to a fraction:
The formula works not only for a single repeating figure, but also for a repeating group of figures. For example:
Note that every series of repeating consecutive decimals can be conveniently simplified with the following:
Archimedes used the sum of a geometric series to compute the area enclosed by a parabola and a straight line. His method was to dissect the area into an infinite number of triangles, as shown in the figure to the right.
Archimedes' Theorem The total area under the parabola is 4/3 of the area of the blue triangle.
Proof: Using his extensive knowledge of geometry, Archimedes determined that each yellow triangle has 1/8 the area of the blue triangle, each green triangle has 1/8 the area of a yellow triangle, and so forth.
Assuming that the blue triangle has area 1, the total area is an infinite sum:
The first term represents the area of the blue triangle, the second term the areas of the two yellow triangles, the third term the areas of the four green triangles, and so on. Simplifying the fractions gives
This is a geometric series with common ratio 1/4 and the fractional part is equal to 1/3:
The sum is
For example, the area inside the Koch snowflake can be described as the union of infinitely many equilateral triangles (see figure). Each side of the green triangle is exactly 1/3 the size of a side of the large blue triangle, and therefore has exactly 1/9 the area. Similarly, each yellow triangle has 1/9 the area of a green triangle, and so forth. Taking the blue triangle as a unit of area, the total area of the snowflake is
The first term of this series represents the area of the blue triangle, the second term the total area of the three green triangles, the third term the total area of the twelve yellow triangles, and so forth. Excluding the initial 1, this series is geometric with constant ratio r = 4/9. The first term of the geometric series is a = 3(1/9) = 1/3, so the sum is
Thus the Koch snowflake has 8/5 of the area of the base triangle.
Understanding the convergence of a geometric series allows to resolve many of Zeno's paradoxes as it reveals that a sum of an infinite set can remain finite for | r | < 1. For example, Zeno's dichotomy paradox maintains that movement is impossible, as one can divide any path into steps of one half of the distance remaining, thus an infinite number of steps is needed to cross any finite distance. The hidden assumption is that a sum of infinite number of finite steps cannot be finite. This is of course not true as evident by the convergence of the geometrical series with r=1/2 illustrated at the picture at the introduction section of this article.
For example, suppose that you expect to receive a payment of $100 once per year in perpetuity. Receiving $100 a year from now is worth less to you than an immediate $100, because you cannot invest the money until you receive it. In particular, the present value of a $100 one year in the future is $100 / (1 + i), where i is the yearly interest rate.
Similarly, a payment of $100 two years in the future has a present value of $100 / (1 + i)2 (squared because it would have received the yearly interest twice). Therefore, the present value of receiving $100 per year in perpetuity can be expressed as an infinite series:
This is a geometric series with common ratio 1 / (1 + i). The sum is
For example, if the yearly interest rate is 10% (i = 0.10), then the entire annuity has a present value of $1000.