Convergence Radii for Eigenvalues of Tri-Diagonal Matrices

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 non-zero entries B(k,k+1) = B(k+1,k) = k(alpha), 0 <=alpha <= 2. The spectrum of L + zB is discrete. For small vertical bar z vertical bar the nth 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