In mathematics, a geometric series is a series
with a constant ratio between successive terms.
For example, the series
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. This relationship
allows for the representation of a geometric
series using only two terms, r and a. The
term r is the common ratio, and a is the first
term of the series. As an example the geometric
series given in the introduction,
may simply be written as
, with and .
The following table shows several geometric
series with different common ratios:
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.
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. The sum of the terms oscillates
between two values. 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 absolute value of the ratio is
less than 1; as the numbers near zero, they
become insignificantly small, allowing a sum
to be calculated despite the series containing
infinitely-many terms. The sum can be computed
using the self-similarity of the series.
Example
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 seriess from the original series s
cancels every term in the original but the
first:
A similar technique can be used to evaluate
any self-similar expression.
Formula
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:
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.
The formula holds true for complex "r", with
the same restrictions.
Proof of convergence
We can prove that the geometric series converges
using the sum formula for a geometric progression:
Since(1−r) = 1−rn+1 and rn+1 → 0 for
| r | < 1.
Convergence of geometric series can also be
demonstrated by rewriting the series as an
equivalent telescoping series. Consider the
function:
Note that:
Thus:
If
then
So S converges to
Generalized formula
For , the sum of the first n terms of a geometric
series is:
where .
We can derive this formula as follows:
we put
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:
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:
That is, a repeating decimal with repeat length
n is equal to the quotient of the repeating
part and 10n - 1.
Archimedes' 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 states that the total
area under the parabola is 4/3 of the area
of the blue triangle.
Archimedes determined that each green triangle
has 1/8 the area of the blue triangle, each
yellow triangle has 1/8 the area of a green
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 green triangles, the third term the
areas of the four yellow 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
The sum is
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
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. 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(13, so the
sum is
Thus the Koch snowflake has 8/5 of the area
of the base triangle.
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 .
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.
For example, suppose that a payment of $100
will be made to the owner of the annuity once
per 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 + 
), where is the yearly interest rate.
Similarly, a payment of $100 two years in
the future has a present value of $100 / (1 + )2.
Therefore, the present value of receiving
$100 per year in perpetuity is
which is the infinite series:
This is a geometric series with common ratio
1 / (1 +  ). The sum is the first term
divided by:
For example, if the yearly interest rate is
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. 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
can be interpreted as a power series in the
Taylor's theorem sense, converging where . From
this, one can extrapolate to obtain other
power series. For example,
By differentiating the geometric series, one
obtains the variant
See also
0.999...
Asymptote
Divergent geometric series
Generalized hypergeometric function
Geometric progression
Neumann series
Ratio test
Root test
Series
Tower of Hanoi
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
External links
Hazewinkel, Michiel, ed., "Geometric progression",
Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 
Weisstein, Eric W., "Geometric Series", MathWorld.
Geometric Series at PlanetMath.org.
Peppard, Kim. "College Algebra Tutorial on
Geometric Sequences and Series". West Texas
A&M University. 
Casselman, Bill. "A Geometric Interpretation
of the Geometric Series". 
"Geometric Series" by Michael Schreiber, Wolfram
Demonstrations Project, 2007.
