Eigenvalues of a certain symmetric tridiagonal Toeplitz matrix

Question

Is there any way that can explicitly calculate eigenvalues (or at least the largest eigenvalue) of the following $n \times n$ symmetric matrix: \begin{pmatrix} 1 & 1 & 0 & 0 & \cdots \\ 1 & 1 & 1 & 0 & \cdots \\ 0 & 1 & 1 & 1 & \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots \end{pmatrix}