Consider a family of infinite tri--diagonal matrices of the form L+zB,
where the matrix L is diagonal with entries Lkk=k2, and the matrix B
is off--diagonal, with nonzero entries Bk,k+1=Bk+1,k=kα,0≤α<2. The spectrum of L+zB is discrete. For small ∣z∣ the
n-th eigenvalue En(z),En(0)=n2, is a well--defined analytic
function. Let Rn 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.$