Suppose I have a symmetric tridiagonal (Jacobi) matrix in the following form:

$ \begin{pmatrix} 1 & a_{1} & 0 & ... & 0 \\\ a_{1} & 1 & a_{2} & & ... \\\ 0 & a_{2} & 1 & ... & 0 \\\ ... & & ... & & a_{n-1} \\\ 0 & ... & 0 & a_{n-1} & 1 \end{pmatrix}, $

where all $0< a_i < 1$ for $i = 1\ldots n-1$ but the matrix is not necessarily diagonally dominant. I am interested in finding a tight upper-bound for the largest eigenvalue of this matrix.

Unfortunately, Eigenvalues of Symmetric Tridiagonal Matrices doesn't have the answer that I am looking for.