  In this paper we compute the closure of the numerical range of certain
periodic tridiagonal operators. This is achieved by showing that the closure of
the numerical range of such operators can be expressed as the closure of the
convex hull of the uncountable union of numerical ranges of certain symbol
matrices. For a special case, this result can be improved so that it is the
convex hull of the union of the numerical ranges of only two matrices. A
conjecture is stated for the general case.
