There are several related series that are known as the binomial series.
The most general is
 |
(1)
|
where 
is a binomial coefficient and 
is a real number. This series converges for 
an integer, or 
(Graham et al. 1994, p. 162). When 
is a positive
integer 
,
the series terminates at 
and can be written in the form
 |
(2)
|
The theorem that any one of these (or several other related forms) holds is known as the binomial theorem.
Special cases give the Taylor series
 |  |  |
(3)
|
 |  |  |
(4)
|
where 
is a Pochhammer symbol and 
. Similarly,
 |  |  |
(5)
|
 |  |  |
(6)
|
which is the so-called negative binomial series.
In particular, the case 
gives
 |  |  |
(7)
|
 |  |  |
(8)
|
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(9)
|
(OEIS A001790 and A046161), where 
is a double factorial and 
is a binomial coefficient.
The binomial series has the continued fraction representation
 |
(10)
|
(Wall 1948, p. 343).
