Geometric series played an important role in the early development of calculus, and continue as a central part of the study of the convergence of series. Geometric series are used throughout mathematics. They have important applications in physics, engineering, biology, economics, computer science, queueing theory, and finance.

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

Repeating Decimal

A repeating decimal can be thought of as a geometric series whose common ratio is a power of $\displaystyle{\frac{1}{10}}$. For example:

$\displaystyle{0.7777 \cdots = \frac{7}{10} + \frac{7}{100}+ \frac{7}{1000}+ \frac{7}{10000}+ \cdots}$

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

$\displaystyle{ \begin{aligned} 0.7777 \cdots &= \frac { a }{ 1-r } \\ &= \frac { \frac{ 7 }{ 10 } }{ 1-\frac{ 1 }{ 10 } } \\ &= \frac{\left(\frac{7}{10}\right)}{\left(\frac{9}{10}\right)} \\ &= \left(\frac{7}{10}\right)\left(\frac{10}{9}\right)\\ &= \frac { 7 }{ 9 } \end{aligned} }$

The formula works for any repeating term. Some more examples are:

$\displaystyle{ \begin{aligned} 0.123412341234 \cdots &= \frac { a }{ 1-r } \\ &= \frac { \frac{1234}{ 10000 } }{ 1-\frac{ 1 }{ 10000 } } \\ &= \frac{\left(\frac{1234}{ 10000 }\right)}{\left(\frac{9999}{10000}\right)} \\ &= \left(\frac{1234}{ 10000 }\right)\left(\frac{10000}{9999}\right)\\ &= \frac { 1234 }{ 9999 } \end{aligned} }$

$\displaystyle{ \begin{aligned} 0.0909090909 \cdots &= \frac { a }{ 1-r } \\ &= \frac { \frac{9}{ 100 } }{ 1-\frac{ 1 }{ 100 } } \\ &= \frac{\left(\frac{9}{ 100 }\right)}{\left(\frac{99}{100}\right)} \\ &= \left(\frac{9}{100}\right)\left(\frac{100}{99}\right)\\ &= \frac { 9 }{ 99 } \\ &= \frac{1}{11} \end{aligned} }$

$\displaystyle{ \begin{aligned} 0.143814381438 \cdots &= \frac { a }{ 1-r } \\ &= \frac { \frac{1438}{ 10000 } }{ 1-\frac{ 1 }{ 10000 } } \\ &= \frac{\left(\frac{1438}{ 10000 }\right)}{\left(\frac{9999}{10000}\right)} \\ &= \left(\frac{1438}{ 10000 }\right)\left(\frac{10000}{9999}\right)\\ &= \frac { 1438 }{ 9999 } \end{aligned} }$

$\displaystyle{ \begin{aligned} 0.9999 \cdots &= \frac { a }{ 1-r } \\ &= \frac { \frac{ 9 }{ 10 } }{ 1-\frac{ 1 }{ 10 } } \\ &= \frac{\left(\frac{9}{10}\right)}{\left(\frac{9}{10}\right)} \\ &= \left(\frac{9}{10}\right)\left(\frac{10}{9}\right)\\ &= \frac { 9 }{ 9 } \\ &= 1 \end{aligned} }$

That is, a repeating decimal with a repeating part of length $n$ is equal to the quotient of the repeating part (as an integer) and $10^n - 1$.

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

Archimedes' dissection of a parabolic segment into infinitely many triangles.

Archimedes' Theorem states that the total area under the parabola is $\displaystyle{\frac{4}{3}}$ of the area of the blue triangle. He determined that each green triangle has $\displaystyle{\frac{1}{8}}$ the area of the blue triangle, each yellow triangle has $\displaystyle{\frac{1}{8}}$ the area of a green triangle, and so forth. Assuming that the blue triangle has area $1$, the total area is an infinite series:

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

The first term represents the area of the blue triangle, the second term the areas of the two green triangles, the third term the areas of the four yellow triangles, and so on. Simplifying the fractions gives:

$\displaystyle{1+ \frac{1}{4}+ \frac{1}{16}+ \frac{1}{64}+ \cdots}$

This is a geometric series with common ratio $\displaystyle{\frac{1}{4}}$, and the fractional part is equal to $\displaystyle{\frac{1}{3}}$.

Fractal Geometry

Koch snowflake

The interior of a Koch snowflake is comprised of an infinite amount of triangles.

The Koch snowflake is a fractal shape with an interior comprised of an infinite amount of triangles. In the study of fractals, geometric series often arise as the perimeter, area, or volume of a self-similar figure. In the case of the Koch snowflake, its area can be described with a geometric series.

Constructing the Koch snowflake: the first four iterations

Each iteration adds a set of triangles to the outside of the shape.

The area inside the Koch snowflake can be described as the union of an infinite number of equilateral triangles. In the diagram above, the triangles added in the second iteration are exactly $\displaystyle{\frac{1}{3}}$ the size of a side of the largest triangle, and therefore they have exactly $\displaystyle{\frac{1}{9}}$ the area. Similarly, each triangle added in the second iteration has $\displaystyle{\frac{1}{9}}$ the area of the triangles added in the previous iteration, and so forth. Taking the first triangle as a unit of area, the total area of the snowflake is:

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

The first term of this series represents the area of the first triangle, the second term the total area of the three triangles added in the second iteration, the third term the total area of the twelve triangles added in the third iteration, and so forth. Excluding the initial term $1$, this series is geometric with constant ratio $\displaystyle{r = \frac{4}{9}}$. The first term of the geometric series is $\displaystyle{a = 3 \frac{1}{9} = \frac{1}{3}}$, so the sum is:

$\displaystyle{ \begin{aligned} 1+ \frac { a }{ 1-r } &=1+ \frac { \frac{1}{3} }{ 1- \frac{4}{9} } \\ &= \frac{8}{5} \end{aligned} }$

Thus the Koch snowflake has $\displaystyle{\frac{8}{5}}$ of the area of the base triangle.

Zeno's Paradoxes

Zeno's Paradoxes are a set of philosophical problems devised by an ancient Greek philosopher to support the doctrine that the truth is contrary to one's senses. Simply stated, one of Zeno's paradoxes says: There is a point, A, that wants to move to another point, B. If A only moves half of the distance between it and point B at a time, it will never get there, because you can continue to divide the remaining space in half forever. Zeno's mistake is in the assumption that the sum of an infinite number of finite steps cannot be finite. We now know that his paradox is not true, as evidenced by the convergence of the geometric series with $\displaystyle{r = \frac{1}{2}}$. This problem has been solved by modern mathematics, which can apply the concept of infinite series to find a sum of the distances traveled.