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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.
i.e., the ratio (or quotient) ''q'' of two consecutive terms is the same for each pair.  


An infinite geometric series converges if and only if |''q''|<1.
Thus, every geometric series has the form
:<math>
a + aq + aq^2 + aq^3 + \cdots
</math>
where the quotient (ratio) of the  (''n''+1)th  and the ''n''th term  is
:<math>
\frac{a q^{n}}{aq^{n-1}} = q.
</math>


Then its sum is <math> a \over 1-q </math> where ''a'' is the first term of the series.
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>).  


== Example ==
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.


The series
In finance, since compound [[interest rate|interest]] generates a geometric sequence,
: <math> 6 + 2 + \frac 2 3 + \frac 2 9 + \frac 2 {27} + \cdots </math>
regular payments together with compound interest lead to a geometric 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 ==
{| class="wikitable"
|-
! Positive ratio
! width=100px | &nbsp;
! Negative ratio
|-
| The series
: <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
: <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>
and first term
and first term
: <math> a = 6 </math>
: <math> a = 6 \,</math>
and therefore its sum is
and therefore its sum is
: <math> { 6 \over 1-\frac 13 } = 9 </math>
: <math> { 6 \over 1-\frac 13 } = 9 </math>
| &nbsp;
| The series
: <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
: <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
: <math> q = - \frac 1 3 </math>
and first term
: <math> a = 6\, </math>
and therefore its sum is
: <math> { 6 \over 1- \left( - \frac 13 \right) } = \frac 9 2 </math>
|}
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>
  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{122}{27}
</math>
== 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 ===


== Power series ==
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''.


Any geometric series  
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  
: <math> \sum_{k=1}^\infty a_k \qquad ( a_k \in \mathbb C ) </math>
: <math> \sum_{k=1}^\infty a_k \qquad ( a_k \in \mathbb C ) </math>
can be written as
can be written as
: <math> a \sum_{k=0}^\infty x^k </math>
: <math> a \sum_{k=0}^\infty q^k </math>
where  
where  
: <math> a = a_1 \qquad \textrm{and} \qquad x = { a_{k+1} \over a_k } \in \mathbb C
: <math> a = a_1 \qquad \textrm{and} \qquad q = { a_{k+1} \over a_k } \in \mathbb C
         \hbox{ is the constant quotient}  
         \hbox{ is the constant quotient}  
   </math>
   </math>


The partial sums of the [[power series]] &Sigma;''x''<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} x^k = 1 + x + x^2 + \cdots + x^{n-1}
       \sum_{k=0}^{n-1} q^k = 1 + q + q^2 + \cdots + q^{n-1}
       =  \begin{cases}
       =  \begin{cases}
                         {\displaystyle \frac{1-x^n}{1-x}} &\hbox{for } x\ne 1 \\
                         {\displaystyle \frac{1-q^n}{1-q}} &\hbox{for } q\ne 1 \\
                                                 n \cdot 1 &\hbox{for } x = 1
                                                 n \cdot 1 &\hbox{for } q = 1
           \end{cases}
           \end{cases}
</math>
</math>
because
because
: <math> (1-x)(1 + x + x^2 + \cdots + x^{n-1}) = 1-x^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-x^n \over 1-x } = {1-\lim_{n\to\infty}x^n \over 1-x } \quad (x\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>
it is
it is
: <math> \lim_{n\to\infty} S_n = {1 \over1-x } \quad \Leftrightarrow \quad |x|<1 </math>
: <math> \lim_{n\to\infty} S_n = {1 \over1-q } \quad \Longleftrightarrow \quad |q|<1 </math>
and the geometric series converges (more precisely: converges absolutely) for |''x''|<1 with the sum
 
: <math> \sum_{k=1}^\infty a_k = { a \over 1-q }</math>
Thus the ''sum'' or ''limit'' of the series is
and diverges for |''x''| &ge; 1.  
: <math> \sum_{k=1}^\infty a_k = { a \over 1-q } \ \text{  for  }\ |q|<1 </math>
(Depending on the sign of ''a'', the limit is +∞ or &minus;∞ for ''x''&ge;1.)
 
== 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 ==
 
In [[combinatorics]], the partial sums of the geometric series are essential for
the definition of [[q-analog|''q''-analogs]], and the following shorthand notation
: <math> [n]_q = 1 + q + q^2 + q^3 + \cdots + q^{n-1} </math>
is used for the ''q''-analogue of a natural number ''n''.[[Category:Suggestion Bot Tag]]

Latest revision as of 06:00, 21 August 2024

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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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a + aq + aq^2 + aq^3 + \cdots }

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{a q^{n}}{aq^{n-1}} = q. }

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \over 1-q } , 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
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6 + 2 + \frac 2 3 + \frac 2 9 + \frac 2 {27} + \cdots + \frac 6 {3^{n-1}} + \cdots }

and corresponding sequence of partial sums

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 }

is a geometric series with quotient

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q = \frac 1 3 }

and first term

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = 6 \,}

and therefore its sum is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle { 6 \over 1-\frac 13 } = 9 }
  The series
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6 - 2 + \frac 2 3 - \frac 2 9 + \frac 2 {27} -+ \cdots (-1)^{n-1}\frac 6 {3^{n-1}} \cdots }

and corresponding sequence of partial sums

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 }

is a geometric series with quotient

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q = - \frac 1 3 }

and first term

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = 6\, }

and therefore its sum is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle { 6 \over 1- \left( - \frac 13 \right) } = \frac 9 2 }

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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} }

and for q = −1/3

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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{122}{27} }

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_n = P \left( 1 + {p\over100} \right)^n }

the second to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_{n-1} = P \left( 1 + {p\over100} \right)^{n-1} }

etc., and the last one to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_1 = P \left( 1 + {p\over100} \right) }

Thus the cumulated sum

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_1+P_2+\cdots P_n = Pq + Pq^2 + \cdots + Pq^n \qquad \text {where } q = 1 + {p\over100} }

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_1 = P \left( {p\over100} \right) } ,

for the payment after two time periods it is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_2 = P \left( {p\over100} \right)^2 } ,

etc., and for the last payment after n time periods the interest is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_n = P \left( {p\over100} \right)^n } .

Thus the accumulated interest

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle nP-L = I_1 +I_2 + \cdots + I_n = Pq + Pq^2 + \cdots + Pq^n \qquad \text {where } q = 1 + {p\over100} }

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{k=1}^\infty a_k \qquad ( a_k \in \mathbb C ) }

can be written as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \sum_{k=0}^\infty q^k }

where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = a_1 \qquad \textrm{and} \qquad q = { a_{k+1} \over a_k } \in \mathbb C \hbox{ is the constant quotient} }

Partial sums

The partial sums of the series Σqk are

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{k=0}^{n-1} q^k = 1 + q + q^2 + \cdots + q^{n-1} = \begin{cases} {\displaystyle \frac{1-q^n}{1-q}} &\hbox{for } q\ne 1 \\ n \cdot 1 &\hbox{for } q = 1 \end{cases} }

because

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1-q)(1 + q + q^2 + \cdots + q^{n-1}) = 1-q^n }

Thus

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 }

Limit

Since

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n\to\infty} {1-q^n \over 1-q } = {1-\lim_{n\to\infty}q^n \over 1-q } \quad (q\ne1)}

it is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n\to\infty} S_n = {1 \over1-q } \quad \Longleftrightarrow \quad |q|<1 }

Thus the sum or limit of the series is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{k=1}^\infty a_k = { a \over 1-q } \ \text{ for }\ |q|<1 }

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):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{k=1}^\infty x^k \ \text{ for }\ x \in \mathbb R \ \text{ or }\ \mathbb C }

The convergence radius of this power series is 1. It

  • converges (more precisely: converges absolutely) for |x|<1 with the sum
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{k=1}^\infty x^k = { 1 \over 1-x }}
  • 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [n]_q = 1 + q + q^2 + q^3 + \cdots + q^{n-1} }

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

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