This is kind of two questions in one.
Firstly, does the following expression have a closed form?
$$\sum_{i=0}^k \binom{n}{i}x^i$$
$$\text{(first $k$ terms in binomial series)}$$
where $k$ is some integer $0 < k < n$.
Secondly, this question got me thinking: is there a way to know if an expression does not have a closed form? If the above expression has no closed form, how would I know?
Best Answer
There is at least an expression.
Write $$\sum_{i=0}^k \binom{n}{i}x^i=\sum_{i=0}^n \binom{n}{i}x^i-\sum_{i=k+1}^n \binom{n}{i}x^i=(1+x)^n-\sum_{i=k+1}^n \binom{n}{i}x^i$$ $$\sum_{i=k+1}^n \binom{n}{i}x^i=x^{k+1} \binom{n}{k+1} \, _2F_1(1,k-n+1;k+2;-x)$$ where appears the Gaussian hypergeometric function.