Geometric series

This article is about infinite geometric series. For finite sums, see geometric progression.

Each of the purple squares is obtained by multiplying the area of the next larger square by 1/4. 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 successive term can be obtained by multiplying the previous term by 1 / 2.

Geometric series are one of the simplest examples of infinite series with finite sums, although not all of them have this property. 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.

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 −1 and +1, the terms of the series become smaller and smaller, approaching zero in the limit and the series converges to a sum. In the case above, where r is one half, 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 in magnitude. 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. See for example Grandi's series: 1 − 1 + 1 − 1 + ···.

Sum

The sum of a geometric series is finite as long as the terms approach zero; as the numbers near zero, they become insignificantly small, allowing a sum to be calculated despite the series being infinite. The sum can be computed using the self-similarity of the series.

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 (2/3)s from the original series cancels every term in the original but the first:

$s \,-\, \tfrac23s \;=\; 1,\;\;\;\;\;\;\;\;\mbox{so }s=3.$

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:

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

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

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

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

$s \;=\; \sum_{k=0}^\infty ar^k = \frac{a}{1-r}=a+ar+ar^2+ar^3+ar^4+\cdots.$

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:

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

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.

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 $\frac{1}{1-r}$.

Applications

Repeating decimals

Main article: Repeating decimal

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:

$0.09090909\ldots \;=\; \frac{09}{99} \;=\; \frac{1}{11}.$

$0.143814381438\ldots \;=\; \frac{1438}{9999}.$

$0.9999\ldots \;=\; \frac{9}{9} \;=\; 1.$

Archimedes' quadrature of the parabola

Main article: The Quadrature of the Parabola

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.

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 a 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

Main article: Zeno's paradoxes

The convergence of a geometric series reveals that a sum involving an infinite number of summands can indeed be finite, and so allows one to resolve many of Zeno's paradoxes. For example, Zeno's dichotomy paradox maintains that movement is impossible, as one can divide any finite path into an infinite number of steps wherein each step is taken to be half the remaining distance. Zeno's mistake is in the assumption that the sum of an infinite number of finite steps cannot be finite. This is of course not true, as evidenced by the convergence of the geometric series with r = 1 / 2.

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

Main article: Time value of money

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 a payment of $100 will be made to the owner of the annuity once per year (at the end of the year) in perpetuity. Receiving $100 a year from now is worth less than an immediate $100, because one cannot invest the money until one receives it. In particular, the present value of $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 two years' worth of interest is lost by not receiving the money right now). Therefore, the present value of receiving $100 per year in perpetuity is

$\sum_{n=1}^\infty \frac{\$100}{(1+I)^n},$

which is the 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 the first term divided by (one minus the common ratio):

$\frac{\$ 100/(1+I)}{1 - 1/(1+I)} \;=\; \frac{\$ 100}{I}.$

For example, if the yearly interest rate is 10% (I = 0.10), then the entire annuity has a present value of $100 / 0.10 = $1000.

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

Geometric power series

The formula for a geometric series

$\frac{1}{1-x}=1+x+x^2+x^3+x^4+\cdots$

can be interpreted as a power series in the Taylor's theorem sense, converging where | x | < 1. From this, one can extrapolate to obtain other power series. For example,

$\tan^{-1}(x)=\int\frac{dx}{1+x^2}=\int\frac{dx}{1-(-x^2)}=\int(1-x^2+x^4-x^6+\cdots)dx=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\cdots$

See also

• asymptote
• series (mathematics)
• geometric progression
• ratio test
• root test
• divergent geometric series
• Neumann series
• Tower of Hanoi
• 0.999...

Specific geometric series

• Grandi's series: 1 − 1 + 1 − 1 + · · ·
• 1 + 2 + 4 + 8 + · · ·
• 1 − 2 + 4 − 8 + · · ·
• 1/2 + 1/4 + 1/8 + 1/16 + · · ·
• 1/2 − 1/4 + 1/8 − 1/16 + · · ·
• 1/4 + 1/16 + 1/64 + 1/256 + · · ·

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 (Mathematical Association of America) 105 (1): 36–40. doi:10.2307/2589524. JSTOR 2589524. 

History and philosophy

• C. H. Edwards, Jr. (1994). The Historical Development of the Calculus, 3rd ed., Springer. ISBN 978-0387943138.
• Swain, Gordon and Thomas Dence (April 1998). "Archimedes' Quadrature of the Parabola Revisited". Mathematics Magazine 71 (2): 123–30. doi:10.2307/2691014. JSTOR 2691014. 
• Eli Maor (1991). To Infinity and Beyond: A Cultural History of the Infinite, Princeton University Press. ISBN 978-0691025117
• Morr Lazerowitz (2000). The Structure of Metaphysics (International Library of Philosophy), Routledge. ISBN 978-0415225267

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

• Weisstein, Eric W., "Geometric Series" from MathWorld.
• Geometric Series at PlanetMath.
• Peppard, Kim. "College Algebra Tutorial on Geometric Sequences and Series". West Texas A&M University. http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut54d_geom.htm. 
• Casselman, Bill. "A Geometric Interpretation of the Geometric Series" (Applet). http://merganser.math.gvsu.edu/calculus/summation/geometric.html. 
• "Geometric Series" by Michael Schreiber, Wolfram Demonstrations Project, 2007.