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In mathematics, a geometric series is a series summing the terms of an infinite geometric sequence, in which the ratio of consecutive terms is constant. For example, the series 1 2 + 1 4 + 1 8 + ⋯ {\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{4}}+{\tfrac {1}{8}}+\cdots } is a geometric series with common ratio 1 2 {\displaystyle {\tfrac {1}{2}}} , which converges to the sum of 1 {\displaystyle 1} . Each term in a geometric series is the geometric mean of the term before it and the term after it, in the same way that each term of an arithmetic series is the arithmetic mean of its neighbors.
is a geometric series with common ratio
1
2
{\displaystyle {\tfrac {1}{2}}}
, which converges to the sum of
1
{\displaystyle 1}
. Each term in a geometric series is the geometric mean of the term before it and the term after it, in the same way that each term of an arithmetic series is the arithmetic mean of its neighbors.
While Greek philosopher Zeno's paradoxes about time and motion (5th century BCE) he been interpreted as involving geometric series, such series were formally studied and applied a century or two later by Greek mathematicians, for example used by Archimedes to calculate the area inside a parabola (3rd century BCE). Today, geometric series are used in mathematical finance, calculating areas of fractals, and various computer science topics.
Though geometric series most commonly involve real or complex numbers, there are also important results and applications for matrix-valued geometric series, function-valued geometric series, p {\displaystyle p} -adic number geometric series, and most generally geometric series of elements of abstract algebraic fields, rings, and semirings.
Definition and examples[edit]
-adic number geometric series, and most generally geometric series of elements of abstract algebraic fields, rings, and semirings.
The geometric series is an infinite series derived from a special type of sequence called a geometric progression. This means that it is the sum of infinitely many terms of geometric progression: starting from the initial term a {\displaystyle a} , and the next one being the initial term multiplied by a constant number known as the common ratio r {\displaystyle r} . By multiplying each term with a common ratio continuously, the geometric series can be defined mathematically as[1] a + a r + a r 2 + a r 3 + ⋯ = ∑ k = 0 ∞ a r k . {\displaystyle a+ar+ar^{2}+ar^{3}+\cdots =\sum _{k=0}^{\infty }ar^{k}.} The sum of a finite initial segment of an infinite geometric series is called a finite geometric series, expressed as[2] a + a r + a r 2 + a r 3 + ⋯ + a r n = ∑ k = 0 n a r k . {\displaystyle a+ar+ar^{2}+ar^{3}+\cdots +ar^{n}=\sum _{k=0}^{n}ar^{k}.}
, and the next one being the initial term multiplied by a constant number known as the common ratio
r
{\displaystyle r}
. By multiplying each term with a common ratio continuously, the geometric series can be defined mathematically as[1]
a
+
a
r
+
a
r
2
+
a
r
3
+
⋯
=
∑
k
=
0
∞
a
r
k
.
{\displaystyle a+ar+ar^{2}+ar^{3}+\cdots =\sum _{k=0}^{\infty }ar^{k}.}
The sum of a finite initial segment of an infinite geometric series is called a finite geometric series, expressed as[2]
a
+
a
r
+
a
r
2
+
a
r
3
+
⋯
+
a
r
n
=
∑
k
=
0
n
a
r
k
.
{\displaystyle a+ar+ar^{2}+ar^{3}+\cdots +ar^{n}=\sum _{k=0}^{n}ar^{k}.}
When r > 1 {\displaystyle r>1} it is often called a growth rate or rate of expansion. When 0
it is often called a growth rate or rate of expansion. When
0