25. Answer. 2
Solution. Let
$$a_{n}=\sum_{i=0}^{n}\binom{n}{i}^{-1}$$

Assume that $n \geq 3$. It is clear that
$$a_{n}=2+\sum_{i=1}^{n-1}\binom{n}{i}^{-1}>2$$

Also note that
$$a_{n}=2+2 / n+\sum_{i=2}^{n-2}\binom{n}{i}^{-1}$$

Since $\binom{n}{i} \geq\binom{ n}{2}$ for all $i$ with $2 \leq i \leq n-2$,
$$a_{n} \leq 2+2 / n+(n-3)\binom{n}{2}^{-1} \leq 2+2 / n+2 / n=2+4 / n .$$

So we have show that for all $n \geq 3$,
$$2<a_{n} \leq 2+4 / n .$$

Thus
$$\lim _{n \rightarrow \infty} a_{n}=2$$