Infinite series involving eigenvalues of the Beltrami-Laplace operator on Riemannian manifolds as well as $L^p$-estimates of eigenfunctions arise in the study of the nonlinear Schrödinger equation (NLS) on compact manifolds. Denote by $\mu_{k} := k(k + 1)$ the eigenvalue of the operator $- \Delta_{\mathbb{S}^2}$ associated to the eigenfunction $e_k \in C^{\infty}(M)$. Now, consider the summation $$ \sum_{k = 0}^{\infty} \frac{1}{ \langle \mu_k - \alpha \rangle \langle \mu_k \rangle^{\varepsilon}}$$ where $\alpha > 0$ (is a positive arbitrary constant), $\varepsilon > 0$ and $\langle x \rangle : = 1 + |x|$. My question is the following:

$$\langle \mu_k - \alpha \rangle^{-1} \langle \mu_k \rangle^{- \varepsilon} \in \ell^{1}_{k}(\mathbb{N}) \mbox{ }, $$

independently of the choice of $ \alpha$?

My failed attempt was to consider two cases: (Case 1) $\mu_k \geq 4 \alpha$. In this case we have $|\mu_k - \alpha| \geq \frac{3}{4} \mu_k$ and one can obtain the desired conclusion. (Case 2) $\mu_k \leq 4 \alpha$. In this case, we have a finite sum, but I would like to prove that the summation is bounded by a constant which does not depends on $\alpha$. Thanks in advance !!!