Editing Binomial coefficient (section)

==== Partial sums ====
Although there is no [[closed formula]] for [[partial sum]]s
: <math> \sum_{j=0}^k \binom n j</math>
of binomial coefficients,<ref>{{citation | last = Boardman | first = Michael | issue = 5 | journal = [[Mathematics Magazine]] | jstor = 3219201 | doi = 10.2307/3219201 | mr = 1573776 | quote = it is well known that there is no closed form (that is, direct formula) for the partial sum of binomial coefficients | pages = 368–372 | title = The Egg-Drop Numbers | volume = 77 | year = 2004}}.</ref> one can again use ({{EquationNote|3}}) and induction to show that for {{math|1=''k'' = 0, …, ''n'' − 1}},
: <math> \sum_{j=0}^k (-1)^j\binom{n}{j} = (-1)^k\binom {n-1}{k},</math>
with special case<ref>see induction developed in eq (7) p. 1389 in {{citation
 | last = Aupetit | first = Michael
 | journal = [[Neurocomputing (journal)|Neurocomputing]]
 | pages = 1379–1389
 | title = Nearly homogeneous multi-partitioning with a deterministic generator
 | volume = 72
 | number = 7–9
 | year = 2009
 | issn = 0925-2312
 | doi = 10.1016/j.neucom.2008.12.024
}}.</ref>

: <math>\sum_{j=0}^n (-1)^j\binom n j = 0</math>
for {{math|''n'' > 0}}. This latter result is also a special case of the result from the theory of [[finite differences]] that for any polynomial ''P''(''x'') of degree less than ''n'',<ref>{{cite journal|last=Ruiz|first=Sebastian|title=An Algebraic Identity Leading to Wilson's Theorem|journal=The Mathematical Gazette|year=1996|volume=80|issue=489|pages=579–582|doi=10.2307/3618534| jstor=3618534|arxiv=math/0406086|s2cid=125556648 }}</ref>
:<math> \sum_{j=0}^n (-1)^j\binom n j P(j) = 0.</math>
Differentiating ({{EquationNote|2}}) ''k'' times and setting ''x'' = −1 yields this for
<math>P(x)=x(x-1)\cdots(x-k+1)</math>,
when 0 ≤ ''k'' < ''n'',
and the general case follows by taking linear combinations of these.

When ''P''(''x'') is of degree less than or equal to ''n'',
{{NumBlk2|:|<math> \sum_{j=0}^n (-1)^j\binom n j P(n-j) = n!a_n</math>|10}}
where <math>a_n</math> is the coefficient of degree ''n'' in ''P''(''x'').

More generally for ({{EquationNote|10}}),
: <math> \sum_{j=0}^n (-1)^j\binom n j P(m+(n-j)d) = d^n n! a_n</math>
where ''m'' and ''d'' are complex numbers. This follows immediately applying ({{EquationNote|10}}) to the polynomial {{tmath|1=Q(x):=P(m + dx)}} instead of {{tmath|P(x)}}, and observing that {{tmath|Q(x)}} still has degree less than or equal to ''n'', and that its coefficient of degree ''n'' is ''d<sup>n</sup>a<sub>n</sub>''.

The [[series (mathematics)|series]] <math display="inline">\frac{k-1}{k} \sum_{j=0}^\infty \frac 1 {\binom {j+x} k}= \frac 1 {\binom{x-1}{k-1}}</math> is convergent for ''k'' ≥ 2. This formula is used in the analysis of the [[German tank problem]]. It follows from <math display="inline">\frac{k-1}k\sum_{j=0}^{M}\frac 1 {\binom{j+x} k}=\frac 1{\binom{x-1}{k-1}}-\frac 1{\binom{M+x}{k-1}}</math> which is proved by [[mathematical induction|induction]] on ''M''.