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A '''geometric series''' is a [[series (mathematics)|series]] associated with a [[geometric sequence]],
A '''geometric series''' is a [[series (mathematics)|series]] associated with a [[geometric sequence]],
i.e., the ratio (or quotient) ''q'' of two consecutive terms is the same for each pair. Thus, the series has the form
i.e., the ratio (or quotient) ''q'' of two consecutive terms is the same for each pair.  
 
Thus, every geometric series has the form
:<math>
:<math>
a + aq + aq^2 + aq^3 + \cdots
a + aq + aq^2 + aq^3 + \cdots
Line 10: Line 12:
  \frac{a q^{n}}{aq^{n-1}} = q.
  \frac{a q^{n}}{aq^{n-1}} = q.
</math>
</math>
The sum of the first ''n'' terms of  a geometric sequence is called the ''n''-th partial sum (of the series); its formula is given below (''S''<sub>''n''</sub>).


An infinite geometric series (i.e., a series with an infinite number of terms)  converges if and only if |''q''|<1, in which case its sum is <math> a \over 1-q </math>, where ''a'' is the first term of the series.
An infinite geometric series (i.e., a series with an infinite number of terms)  converges if and only if |''q''|<1, in which case its sum is <math> a \over 1-q </math>, where ''a'' is the first term of the series.


'''Remarks''' <br>
In finance, since compound [[interest rate|interest]] generates a geometric sequence,
# The sum of finite  (''n'') terms of  a geometric sequence is a finite number ''S''<sub>''n''</sub>;  its formula is given below.
regular payments together with compound interest lead to a geometric series.
#Since every finite geometric sequence is the initial segment of a uniquely determined infinite geometric sequence every finite geometric series is the initial segment of a corresponding infinite geometric series. Therefore, while in elementary mathematics the difference between "finite" and "infinite" may be stressed, in more advanced mathematical texts "geometrical series" usually refers to the infinite series.
 
'''Remark''' <br>
Since every finite geometric sequence is the initial segment of a uniquely determined infinite geometric sequence every finite geometric series is the initial segment of a corresponding infinite geometric series. Therefore, while in elementary mathematics the difference between "finite" and "infinite" may be stressed, in more advanced mathematical texts "geometrical series" usually refers to the infinite series.


== Examples ==
== Examples ==
Line 25: Line 31:
|-
|-
| The series
| The series
: <math> 6 + 2 + \frac 2 3 + \frac 2 9 + \frac 2 {27} + \cdots </math>
: <math> 6 + 2 + \frac 2 3 + \frac 2 9 + \frac 2 {27} + \cdots + \frac 6 {3^{n-1}} + \cdots </math>
and corresponding sequence of partial sums
and corresponding sequence of partial sums
: <math> 6 , 8 , \frac {26} 3 , \frac {80} 9 , \frac {242} {27} , \cdots </math>
: <math> 6 , 8 , \frac {26} 3 , \frac {80} 9 , \frac {242} {27} , \dots , 
          6 \cdot { 1 - \left( \frac 13 \right)^n \over 1- \frac 13 } , \dots </math>
is a geometric series with quotient  
is a geometric series with quotient  
: <math> q = \frac 1 3 </math>
: <math> q = \frac 1 3 </math>
Line 36: Line 43:
| &nbsp;
| &nbsp;
| The series
| The series
: <math> 6 - 2 + \frac 2 3 - \frac 2 9 + \frac 2 {27} -+ \cdots </math>
: <math> 6 - 2 + \frac 2 3 - \frac 2 9 + \frac 2 {27} -+ \cdots (-1)^{n-1}\frac 6 {3^{n-1}} \cdots </math>
and corresponding sequence of partial sums
and corresponding sequence of partial sums
: <math> 6 , 4 , \frac {14} 3 , \frac {40} 9 , \frac {122} {27} , \cdots </math>
: <math> 6 , 4 , \frac {14} 3 , \frac {40} 9 , \frac {122} {27} , \dots ,
        6 \cdot { 1 - \left( - \frac 13 \right)^n \over 1- \frac 13 } , \dots </math>
is a geometric series with quotient  
is a geometric series with quotient  
: <math> q = - \frac 1 3 </math>
: <math> q = - \frac 1 3 </math>
Line 47: Line 55:
|}
|}


The partial sum ''S''<sub>5</sub> follows thus (see the formula derived below)
The sum of the first 5 terms &mdash; the partial sum ''S''<sub>5</sub> (see the formula derived below) &mdash;
is for ''q'' = 1/3
:<math>
  S_5 = 6 + 2 + \frac 2 3 + \frac 2 9 + \frac 2 {27}
      = 6 \left[ 1+\frac{1}{3} + \Big(\frac{1}{3}\Big)^2 + \Big(\frac{1}{3}\Big)^3 + \Big(\frac{1}{3}\Big)^4 \right]
      = 6 \left[ \frac{1-(\frac{1}{3})^5 }{ 1-\frac{1}{3} } \right]
      = \frac{242}{27}
</math>
and for ''q'' = &minus;1/3
:<math>
:<math>
S_5 \equiv 6 + 2 + \frac 2 3 + \frac 2 9 + \frac 2 {27} = 6 \left[ 1+\frac{1}{3} + \Big(\frac{1}{3}\Big)^2
  S_5 = 6 - 2 + \frac 2 3 - \frac 2 9 + \frac 2 {27}  
+\Big(\frac{1}{3}\Big)^3 +\Big(\frac{1}{3}\Big)^4 \right]  = 6\left[ \frac{1-(\frac{1}{3})^5}{1-\frac{1}{3}} \right]
      = 6 \left[ 1-\frac{1}{3} + \Big(\frac{1}{3}\Big)^2 - \Big(\frac{1}{3}\Big)^3 + \Big(\frac{1}{3}\Big)^4 \right]   
= \frac{242}{27}
      = 6 \left[ \frac{ 1+(\frac{1}{3})^5 }{ 1+\frac{1}{3} } \right]
      = \frac{122}{27}
</math>
</math>


== Power series ==
== Application in finance ==
 
When regular payments are combined with compound interest this generates a geometric series:
 
=== Regular deposits ===
 
If, for ''n'' time periods, a sum ''P'' is deposited at an interest rate of ''p'' percent,
then &mdash; after the ''n''-th period &mdash;
 
the first payment has increased to
<math> P_n = P \left( 1 + {p\over100} \right)^n </math>
 
the second to
<math> P_{n-1} = P \left( 1 + {p\over100} \right)^{n-1} </math>
 
etc., and the last one to
<math> P_1 = P \left( 1 + {p\over100} \right) </math>
 
Thus the cumulated sum
: <math>  P_1+P_2+\cdots P_n = Pq + Pq^2 + \cdots + Pq^n \qquad
      \text {where } q = 1 + {p\over100}
</math>
is the ''n''-th partial sum of a geometric series.
 
=== Regular down payments ===
 
If a loan ''L'' is to be payed off by ''n'' regular payments ''P'',
the total payment ''nP'' has to cover both the loan ''L'' and the accumulated interest ''I''.
 
The interest for the payment at the end of the first time period is
<math> I_1 = P \left( {p\over100} \right) </math>,
 
for the payment after two time periods it is
<math> I_2 = P \left( {p\over100} \right)^2 </math>,
 
etc., and for the last payment after ''n'' time periods the interest is
<math> I_n = P \left( {p\over100} \right)^n </math>.
 
Thus the accumulated interest
: <math> nP-L = I_1 +I_2 + \cdots + I_n = Pq + Pq^2 + \cdots + Pq^n \qquad
      \text {where } q = 1 + {p\over100}
  </math>
is the ''n''-th partial sum of a geometric series.
(From this equation, ''P'' can easily be calculated.)
 
== Mathematical treatment ==


By definition, a  geometric series  
By definition, a  geometric series  
Line 65: Line 127:
   </math>
   </math>


The partial sums of the [[power series]] &Sigma;''q''<sup>''k''</sup> are
=== Partial sums ===
 
The partial sums of the series &Sigma;''q''<sup>''k''</sup> are
: <math>
: <math>
       S_n = \sum_{k=0}^{n-1} q^k = 1 + q + q^2 + \cdots + q^{n-1}
       \sum_{k=0}^{n-1} q^k = 1 + q + q^2 + \cdots + q^{n-1}
       =  \begin{cases}
       =  \begin{cases}
                         {\displaystyle \frac{1-q^n}{1-q}} &\hbox{for } q\ne 1 \\
                         {\displaystyle \frac{1-q^n}{1-q}} &\hbox{for } q\ne 1 \\
Line 75: Line 139:
because
because
: <math> (1-q)(1 + q + q^2 + \cdots + q^{n-1}) = 1-q^n </math>
: <math> (1-q)(1 + q + q^2 + \cdots + q^{n-1}) = 1-q^n </math>
Thus
: <math> S_n = \sum_{k=1}^n a_k = a\frac{1-q^n}{1-q} \text{ for } q \ne 1 \text{ and } S_n = an \text{ for } q=1 </math>
=== Limit ===
Since  
Since  
: <math> \lim_{n\to\infty} {1-q^n \over 1-q } = {1-\lim_{n\to\infty}q^n \over 1-q } \quad (q\ne1)</math>
: <math> \lim_{n\to\infty} {1-q^n \over 1-q } = {1-\lim_{n\to\infty}q^n \over 1-q } \quad (q\ne1)</math>
Line 80: Line 150:
: <math> \lim_{n\to\infty} S_n = {1 \over1-q } \quad \Longleftrightarrow \quad |q|<1 </math>
: <math> \lim_{n\to\infty} S_n = {1 \over1-q } \quad \Longleftrightarrow \quad |q|<1 </math>


=== Convergence behaviour ===
Thus the ''sum'' or ''limit'' of the series is
: <math> \sum_{k=1}^\infty a_k = { a \over 1-q } \ \text{  for  }\ |q|<1 </math>
 
== Geometric power series ==
 
For each ''q'', the geometric series is a series of numbers, but
since &mdash; apart from the constant factor ''a'' &mdash; they all have the same form &Sigma;''q''<sup>''k''</sup>,
it is convenient to replace the quotient ''q'' by a variable ''x'' and consider the (real or complex) geometric [[power series]]
(a series of functions):
 
:: <math> \sum_{k=1}^\infty x^k \ \text{ for }\ x \in \mathbb R \ \text{ or }\ \mathbb C </math>
 
The [[convergence radius]] of this power series is 1. It
* converges (more precisely: converges [[absolute convergent|absolutely]]) for |''x''|<1 with the sum
:: <math> \sum_{k=1}^\infty x^k = { 1 \over 1-x }</math>
* and diverges for |''x''| &ge; 1.
:* For real ''x'':
:: For ''x'' &ge; 1 the limit is +∞.
:: For ''x'' = &minus;1 the series alternates between 1 and 0.
:: For ''x'' < &minus;1 the sign of partial sums alternates, the limit of their absolute values is ∞, but no infinite limit exists.
:* For complex ''x'':
:: For |''x''| = 1 and ''x'' ≠ 1 (i.e., ''x'' = &minus;1 or non-real complex) the partial sums ''S''<sub>n</sub> are bounded but not convergent.
:: For |''x''| > 1 and ''x'' non-real complex the partial sums oscillate, the limit of their absolute values is ∞, but no infinite limit exists.
 
== A notation: ''q''-analogues ==


The geometric series  
In [[combinatorics]], the partial sums of the geometric series are essential for
* converges (more precisely: converges [[absolute convergent|absolutely]]) for |''q''|<1 with the sum
the definition of [[q-analog|''q''-analogs]], and the following shorthand notation
:: <math> \sum_{k=1}^\infty a_k = { a \over 1-q }</math>
: <math> [n]_q = 1 + q + q^2 + q^3 + \cdots + q^{n-1} </math>
* and diverges for |''q''| &ge; 1.
is used for the ''q''-analogue of a natural number ''n''.
** <math> \lim |S_n| = \infty </math> for |''q''| > 1, and
** Depending on the sign of ''a'', the limit is +∞ or &minus;∞ for ''q''&ge;1.
** In the case ''q''&le;-1 partial sums oscillate and hence no limit exists.)

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A geometric series is a series associated with a geometric sequence, i.e., the ratio (or quotient) q of two consecutive terms is the same for each pair.

Thus, every geometric series has the form

where the quotient (ratio) of the (n+1)th and the nth term is

The sum of the first n terms of a geometric sequence is called the n-th partial sum (of the series); its formula is given below (Sn).

An infinite geometric series (i.e., a series with an infinite number of terms) converges if and only if |q|<1, in which case its sum is , where a is the first term of the series.

In finance, since compound interest generates a geometric sequence, regular payments together with compound interest lead to a geometric series.

Remark
Since every finite geometric sequence is the initial segment of a uniquely determined infinite geometric sequence every finite geometric series is the initial segment of a corresponding infinite geometric series. Therefore, while in elementary mathematics the difference between "finite" and "infinite" may be stressed, in more advanced mathematical texts "geometrical series" usually refers to the infinite series.

Examples

Positive ratio   Negative ratio
The series

and corresponding sequence of partial sums

is a geometric series with quotient

and first term

and therefore its sum is

  The series

and corresponding sequence of partial sums

is a geometric series with quotient

and first term

and therefore its sum is

The sum of the first 5 terms — the partial sum S5 (see the formula derived below) — is for q = 1/3

and for q = −1/3

Application in finance

When regular payments are combined with compound interest this generates a geometric series:

Regular deposits

If, for n time periods, a sum P is deposited at an interest rate of p percent, then — after the n-th period —

the first payment has increased to

the second to

etc., and the last one to

Thus the cumulated sum

is the n-th partial sum of a geometric series.

Regular down payments

If a loan L is to be payed off by n regular payments P, the total payment nP has to cover both the loan L and the accumulated interest I.

The interest for the payment at the end of the first time period is ,

for the payment after two time periods it is ,

etc., and for the last payment after n time periods the interest is .

Thus the accumulated interest

is the n-th partial sum of a geometric series. (From this equation, P can easily be calculated.)

Mathematical treatment

By definition, a geometric series

can be written as

where

Partial sums

The partial sums of the series Σqk are

because

Thus

Limit

Since

it is

Thus the sum or limit of the series is

Geometric power series

For each q, the geometric series is a series of numbers, but since — apart from the constant factor a — they all have the same form Σqk, it is convenient to replace the quotient q by a variable x and consider the (real or complex) geometric power series (a series of functions):

The convergence radius of this power series is 1. It

  • converges (more precisely: converges absolutely) for |x|<1 with the sum
  • and diverges for |x| ≥ 1.
  • For real x:
For x ≥ 1 the limit is +∞.
For x = −1 the series alternates between 1 and 0.
For x < −1 the sign of partial sums alternates, the limit of their absolute values is ∞, but no infinite limit exists.
  • For complex x:
For |x| = 1 and x ≠ 1 (i.e., x = −1 or non-real complex) the partial sums Sn are bounded but not convergent.
For |x| > 1 and x non-real complex the partial sums oscillate, the limit of their absolute values is ∞, but no infinite limit exists.

A notation: q-analogues

In combinatorics, the partial sums of the geometric series are essential for the definition of q-analogs, and the following shorthand notation

is used for the q-analogue of a natural number n.