Let \(\mathcal{H}\) be a separable Hilbert space and let \((e_{n})\) be an orthonormal basis of \(\mathcal{H}\). Let \(T\) be a given selfadjoint operator in \(\mathcal{H}\), denote by \(P_{N}\) the projection operator of \(\mathcal{H}\) onto the subspace \(\mathcal{H}_{N}\) spanned by the elements \(\{e_{1},\dots,e_{n}\}\), and consider the truncated operator \(T_{N}=P_{N}TP_{N}\). Denote by \(\Lambda(T)\) the set of all limit points of eigenvalues of \(T_{N}\) as \(N\rightarrow\infty\). For the case of a tridiagonal operator \(T\), i.e., \[ Te_{n}=a_{n}e_{n+1}+a_{n-1}e_{n-1}+b_{n}e_{n}\quad(n=1,2,\dots), \] the authors give conditions under which the spectrum \(\sigma(T)\) of \(T\) coincides with the set \(\Lambda (T)\). In particular, they prove that if \(T\) is a selfadjoint operator with the property \(\sigma (T) = \Lambda (T)\), then the continued fraction associated with \(T\)
\[ K(\lambda)=\frac{1}{\lambda-b_1-\frac{a_{1}^{2}}{\lambda-b_2-\frac{a_{2}^{2}}{\lambda - b_3-\dots}}} \] represents a meromorphic function in \(\mathbb{C}\) (resp., \(\mathbb{C}\backslash\{0\}\)) if \(T\) has a discrete spectrum with eigenvalues \(\lambda_{n}\) such that \(\lim_{n \rightarrow \infty}|\lambda_{n}|=\infty\) (resp., \(\lim_{n\rightarrow\infty}|\lambda_{n}|=0\)).