<s>[INST] Estimating the limit $\lim_{n\to\infty}\frac{n _2F_1[1 - n, 1 + 2^n n; 2 + 2^n n; -1]}{1 + 2^n n}$ Can you please help me solve the limit below? 
$$\lim_{n\to\infty}\frac{n _2F_1[1 - n, 1 + 2^n n; 2 + 2^n n; -1]}{1 + 
    2^n n}$$ 
where $_2F_1(a,b;c;z)$ - hypergeometric function
 [/INST] The hypergeometric function is a polynomial:
\begin{eqnarray*}
& & {}_2F_1[1 - n, 1 + 2^n n; 2 + 2^n n;z]\\
&=&\sum_{k\ge 0} \frac{(1-n)_k (1+2^n n)_k}{(2+2^n n)_k k!} z^k\\
& =& \sum_{0\le k\le n-1} \frac{(-1)^k(n-1)\cdots (n-k) (2^n n+1)}{(2^n n + k + 1) k!} z^k\\
&=& \sum_{0\le k\le n-1} \binom{n-1}{k} \left(1-\frac{k}{2^n n + k + 1}\right)(-z)^k
\end{eqnarray*}
so
\begin{eqnarray*}
&&
n {}_2F_1[1 - n, 1 + 2^n n; 2 + 2^n n;-1]\\
&=& n \sum_{0\le k\le n-1} \binom{n-1}{k} \left(1-\frac{k}{2^n n + k + 1}\right)\\
&=&  2^{n-1}n - n \sum_{0\le k\le n-1} \frac{k}{2^n n + k + 1}\binom{n-1}{k}\\
&=&  2^{n-1}n + O(n),
\end{eqnarray*}
meaning that the limit is $\frac 12 $.
</s>