Geometric series: Wikis

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The sum of the areas of the purple squares is one third of the area of the large square.

In mathematics, a geometric series is a series with a constant ratio between successive terms. For example, the series

\frac{1}{2} \,+\, \frac{1}{4} \,+\, \frac{1}{8} \,+\, \frac{1}{16} \,+\, \cdots

is geometric, because each term except the first can be obtained by multiplying the previous term by \frac{1}{2} \ .

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.

Contents

Common ratio

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:

Common ratio Example
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:

If r is between minus 1 and plus 1 the terms of the series become smaller and smaller, approaching zero in the limit. The series converges to a sum, as in the case above, where r is a half, and the series has the sum one.
If r is greater than one or less than minus one the terms of the series become larger and larger. The sum of the terms also gets larger and larger, and the series has no sum. (The series diverges.)
If r is equal to one, all of the terms of the series are the same. The series diverges.
If r is minus one the terms take two values alternately (e.g. 2, -2, 2, -2, 2,... ). The sum of the terms oscillates between two values (e.g. 2, 0, 2, 0, 2,... ). This is a different type of divergence and again the series has no sum.

Sum

The sum of a geometric series is finite as long as the terms approach zero. The sum can be computed using the self-similarity of the series.

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Example

A self-similar illustration of the sum s. Removing the largest circle results in a similar figure of 2/3 the original size.

Consider the sum of the following geometric series:

s \;=\; 1 \,+\, \frac{2}{3} \,+\, \frac{4}{9} \,+\, \frac{8}{27} \,+\, \cdots

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:

\frac{2}{3}s \;=\; \frac{2}{3} \,+\, \frac{4}{9} \,+\, \frac{8}{27} \,+\, \frac{16}{81} \,+\, \cdots

This new series is the same as the original, except that the first term is missing. Subtracting the new series \frac{2}{3}s from the original series cancels every term in the original but the first:

s \,-\, \frac{2}{3}s \;=\; 1,\;\;\;\;\;\;\;\;\mbox{so }s=3.

A similar technique can be used to evaluate any self-similar expression.

Formula

For r\neq 1, the sum of the first n terms of a geometric series is:

a + ar + a r^2 + ar^3 + \cdots + a r^{n} = \sum_{k=0}^{n} ar^k= a \, \frac{1-r^{n+1}}{1-r},

where a is the first term of the series, and r is the common ratio. We can derive this formula as follows:

 \begin{align} &\text{Let }s = 1 + r + r^2 + r^3 + \cdots + r^{n}. \\[4pt] &\text{Then }rs = r + r^2 + r^3 + r^4 + \cdots + r^{n} + r^{n+1}. \\[4pt] &\text{Then }s - rs = s(1-r) = 1-r^{n+1},\text{ so }s = \frac{1-r^{n+1}}{1-r}. \end{align}

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

s \;=\; \sum_{k=0}^\infty ar^k = \frac{a}{1-r}.

When a = 1, this simplifies to:

1 \,+\, r \,+\, r^2 \,+\, r^3 \,+\, \cdots \;=\; \frac{1}{1-r},

the left-hand side being a geometric series with common ratio r. We can derive this formula:

 \begin{align} &\text{Let }s = 1 + r + r^2 + r^3 + \cdots. \\[4pt] &\text{Then }rs = r + r^2 + r^3 + r^4 + \cdots. \\[4pt] &\text{Then }s - rs = 1,\text{ so }s(1 - r) = 1,\text{ and thus }s = \frac{1}{1-r}. \end{align}

The general formula follows if we multiply through by a.

This formula is only valid for convergent series (i.e., when the magnitude of r is less than one). For example, the sum is undefined when r = 10, even though the formula gives s = −1/9.

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:

Geometric-view-of-geometric-series.png

Proof of convergence

We can prove that the geometric series converges using the sum formula for a geometric progression:

\begin{align} &1 \,+\, r \,+\, r^2 \,+\, r^3 \,+\, \cdots \\[3pt] &=\; \lim_{n\rightarrow\infty} \left(1 \,+\, r \,+\, r^2 \,+\, \cdots \,+\, r^n\right) \ &=\; \lim_{n\rightarrow\infty} \frac{1-r^{n+1}}{1-r} \end{align}

Since (1 + r + r2 + ... + rn)(1 - r) = 1-rn+1 and rn+1 → 0 for | r | < 1, the limit is 1 /(1 − r).

Applications

Repeating decimals

A repeating decimal can be thought of as a geometric series whose common ratio is a power of 1/10. For example:

0.7777\ldots \;=\; \frac{7}{10} \,+\, \frac{7}{100} \,+\, \frac{7}{1000} \,+\, \frac{7}{10,000} \,+\, \cdots.

The formula for the sum of a geometric series can be used to convert the decimal to a fraction:

0.7777\ldots \;=\; \frac{a}{1-r} \;=\; \frac{7/10}{1-1/10} \;=\; \frac{7}{9}.

The formula works not only for a single repeating figure, but also for a repeating group of figures. For example:

0.123412341234\ldots \;=\; \frac{a}{1-r} \;=\; \frac{1234/10000}{1-1/10000} \;=\; \frac{1234}{9999}.

Note that every series of repeating consecutive decimals can be conveniently simplified with the following:

0.09090909\ldots \;=\; \frac{09}{99} \;=\; \frac{1}{11}.
0.143814381438\ldots \;=\; \frac{1438}{9999}.

Archimedes' quadrature of the parabola

The area enclosed by a parabola and a line is the union of infinitely many triangles.

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:

1 \,+\, 2\left(\frac{1}{8}\right) \,+\, 4\left(\frac{1}{8}\right)^2 \,+\, 8\left(\frac{1}{8}\right)^3 \,+\, \cdots.

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

1 \,+\, \frac{1}{4} \,+\, \frac{1}{16} \,+\, \frac{1}{64} \,+\, \cdots.

This is a geometric series with common ratio 1/4 and the fractional part is equal to 1/3:

\sum_{n=0}^\infty 4^{-n} = 1 + 4^{-1} + 4^{-2} + 4^{-3} + \cdots = {4\over 3}. \;

The sum is

\frac{1}{1 -r}\;=\;\frac{1}{1 -\frac{1}{4}}\;=\;\frac{4}{3}.    Q.E.D.

This computation uses the method of exhaustion, an early version of integration. In modern calculus, the same area could be found using a definite integral.

Fractal geometry

The interior of the Koch snowflake is the union of infinitely many triangles.

In the study of fractals, geometric series often arise as the perimeter, area, or volume of a self-similar figure.

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

1 \,+\, 3\left(\frac{1}{9}\right) \,+\, 12\left(\frac{1}{9}\right)^2 \,+\, 48\left(\frac{1}{9}\right)^3 \,+\, \cdots.

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

1\,+\,\frac{a}{1-r}\;=\;1\,+\,\frac{\frac{1}{3}}{1-\frac{4}{9}}\;=\;\frac{8}{5}.

Thus the Koch snowflake has 8/5 of the area of the base triangle.

Zeno's paradoxes

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.

Euclid

Book IX, Proposition 35 of Euclid's Elements expresses the partial sum of a geometric series in terms of members of the series. It is equivalent to the modern formula.

Economics

In economics, geometric series are used to represent the present value of an annuity (a sum of money to be paid in regular intervals).

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:

\frac{\$ 100}{1+i} \,+\, \frac{\$ 100}{(1+i)^2} \,+\, \frac{\$ 100}{(1+i)^3} \,+\, \frac{\$ 100}{(1+i)^4} \,+\, \cdots.

This is a geometric series with common ratio 1 / (1 + i). The sum is

\frac{a}{1-r} \;=\; \frac{\$ 100/(1+i)}{1 - 1/(1+i)} \;=\; \frac{\$ 100}{i}.

For example, if the yearly interest rate is 10% (i = 0.10), then the entire annuity has a present value of $1000.

This sort of calculation is used to compute the APR of a loan (such as a mortgage). It can also be used to estimate the present value of expected stock dividends, or the terminal value of a security.

Geometric power series

See also

Specific geometric series

References

  • James Stewart (2002). Calculus, 5th ed., Brooks Cole. ISBN 978-0534393397
  • Larson, Hostetler, and Edwards (2005). Calculus with Analytic Geometry, 8th ed., Houghton Mifflin Company. ISBN 978-0618502981
  • Roger B. Nelsen (1997). Proofs without Words: Exercises in Visual Thinking, The Mathematical Association of America. ISBN 978-0883857007
  • Andrews, George E. (1998). "The geometric series in calculus". The American Mathematical Monthly 105 (1): 36–40. doi:10.2307/2589524. 

History and philosophy

Economics

  • Carl P. Simon and Lawrence Blume (1994). Mathematics for Economists, W. W. Norton & Company. ISBN 978-0393957334
  • Mike Rosser (2003). Basic Mathematics for Economists, 2nd ed., Routledge. ISBN 978-0415267847

Biology

  • Edward Batschelet (1992). Introduction to Mathematics for Life Scientists, 3rd ed., Springer. ISBN 978-0387096483
  • Richard F. Burton (1998). Biology by Numbers: An Encouragement to Quantitative Thinking, Cambridge University Press. ISBN 978-0521576987

Computer science

  • John Rast Hubbard (2000). Schaum's Outline of Theory and Problems of Data Structures With Java, McGraw-Hill. ISBN 978-0071378703

External links


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