  Consider a family of infinite tri--diagonal matrices of the form $L+ zB,$
where the matrix $L$ is diagonal with entries $L_{kk}= k^2,$ and the matrix $B$
is off--diagonal, with nonzero entries $B_{k,{k+1}}=B_{{k+1},k}= k^\alpha, 0
\leq \alpha < 2.$ The spectrum of $L+ zB$ is discrete. For small $|z|$ the
$n$-th eigenvalue $E_n (z), E_n (0) = n^2,$ is a well--defined analytic
function. Let $R_n$ be the convergence radius of its Taylor's series about $z=
0.$ It is proved that $$ R_n \leq C(\alpha) n^{2-\alpha} \quad \text{if} 0 \leq
\alpha <11/6.$$
