structural:scholar_edition
Partial sums

Although there is no closed formula for partial sums

of binomial coefficients, one can again use (3) and induction to show that for k = 0, …, n − 1,

with special case

for 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,

Differentiating (2) k times and setting x = −1 yields this for P (x) = x (x − 1) ⋯ (x − k + 1) {\displaystyle P(x)=x(x-1)\cdots (x-k+1)},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,

where a n {\displaystyle a_{n}} is the coefficient of degree n in P (x).

More generally for (10),

where m and d are complex numbers. This follows immediately applying (10) to the polynomial Q (x):= P (m + d x) {\displaystyle Q(x):=P(m+dx)} instead of P (x) {\displaystyle P(x)}, and observing that Q (x) {\displaystyle Q(x)} still has degree less than or equal to n, and that its coefficient of degree n is d a.

The series k − 1 k ∑ j = 0 ∞ 1 (j + x k) = 1 (x − 1 k − 1) {\textstyle {\frac {k-1}{k}}\sum _{j=0}^{\infty }{\frac {1}{\binom {j+x}{k}}}={\frac {1}{\binom {x-1}{k-1}}}} is convergent for k ≥ 2. This formula is used in the analysis of the German tank problem. It follows from k − 1 k ∑ j = 0 M 1 (j + x k) = 1 (x − 1 k − 1) − 1 (M + x k − 1) {\textstyle {\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}}}} which is proved by induction on M.