  We have performed high-precision calculations of the hyperfine structure for
n 2S_1/2 and n 2P_1/2 states of the alkali-metal atoms Rb, Cs, and Fr across
principal quantum number n, and studied the trend in the size of the
correlations. Our calculations were performed in the all-orders correlation
potential method. We demonstrate that the relative correlation corrections fall
off quickly with n and tend towards constant and non-zero values for
highly-excited states. This trend is supported by experiment, and we utilize
the smooth dependence on n to make high-accuracy predictions of the hyperfine
constants, with uncertainties to within 0.1% for most states of Rb and Cs.
