Solution
= Solution
The <binomial theorem> gives
$$
(1+0.01)^9=1+9(0.01)+36(0.01)^2+84(0.01)^3+126(0.01)^4+\cdots
=1.093685272\ldots.
$$
Thus, to three decimal places,
$$
\boxed{(1.01)^9\simeq1.094.}
$$
The first three nonconstant terms already determine this rounding; the remaining positive terms total less than $1.3\times10^{-6}$.