This post is just to help a friend that wanted to see why

\displaystyle \sum_{i=1}^{\lfloor\sqrt{n}\rfloor} \binom{n}{i}

was superpolynomial.

Getting a simple lower bound

We can start by keeping only the last term. As all the terms are positive, that will give us a lower bound:

\displaystyle \sum_{i=1}^{\lfloor\sqrt{n}\rfloor} \binom{n}{i} \ge \binom{n}{\lfloor\sqrt{n}\rfloor}.

We can expand the binomial coefficient in terms of factorials

\displaystyle \binom{n}{\lfloor\sqrt{n}\rfloor} = \frac{n!}{(n - \lfloor\sqrt{n}\rfloor)!\lfloor\sqrt{n}\rfloor!},

and some of the factors inside n! can then be cancelled with the ones in (n - \lfloor\sqrt{n}\rfloor)!, leaving the following expression:

\displaystyle \binom{n}{\lfloor\sqrt{n}\rfloor} = \frac{n(n-1)\hdots(n - \lfloor\sqrt{n}\rfloor + 1)}{\lfloor\sqrt{n}\rfloor!}.

As we want a lower bound, we can bound the numerator by below and the denominator by above, leaving us a bound without factorials:

\displaystyle \frac{n(n-1)\hdots(n - \lfloor\sqrt{n}\rfloor + 1)}{\lfloor\sqrt{n}\rfloor!} \ge \frac{(n - \sqrt{n})^{\sqrt{n} - 1}}{\sqrt{n}^{\sqrt{n}}}.

For any reasonably big value of n we will have n - \lfloor\sqrt{n}\rfloor \ge n/2. If we combine that with factoring out the -1 in the exponent we get

\displaystyle \frac{(n - \sqrt{n})^{\sqrt{n} - 1}}{\sqrt{n}^{\sqrt{n}}} \ge \frac{1}{n - \sqrt{n}}\left(\frac{n}{\sqrt{n}}\right)^{\sqrt{n}} \ge \frac{1}{n}\sqrt{n}^{\sqrt{n}},

removing the problematic floors and arriving to a much simpler expression.

Combining all these bounds we get

\displaystyle \sum_{i=1}^{\lfloor\sqrt{n}\rfloor} \binom{n}{i} \ge \frac{1}{n}\sqrt{n}^{\sqrt{n}},

a much simpler lower bound when compared to the original expression.

Proving it is superpolynomial

If w take an arbitrary polynomial p(n) of degree k we can say for n \ge 1 that

\displaystyle p(n) = \sum_{i=1}^k p_i\,n^i \le \left(\sum_{i=1}^k |p_i|\right)n^k,

meaning we can bound by above any polynomial of degree k by a polynomial of the same degree with a single term.

Taking the limit of the quotient of logarithms,

\displaystyle \lim_{x \to +\infty} \frac{\log \frac{1}{n}\sqrt{n}^{\sqrt{n}}}{\log a\,n^k} = \lim_{x \to +\infty} \frac{\frac{\sqrt{n}}{2}\log n - \log n}{\log a + k\,\log n}

\displaystyle = \lim_{x \to +\infty} \frac{\log n\left(\frac{\sqrt{n}}{2} - 1\right)}{\log n\left(\frac{\log a}{\log n} + k\right)}

\displaystyle = \lim_{x \to +\infty} \frac{\frac{\sqrt{n}}{2} - 1}{\frac{\log a}{\log n} + k}

\displaystyle = +\infty

we can see the lower bound of the original expression is superpolynomial, making the original expression superpolynomial too.