  The classical Mercer's theorem claims that a continuous positive definite
kernel $K({\mathbf x}, {\mathbf y})$ on a compact set can be represented as
$\sum_{i=1}^\infty \lambda_i\phi_i({\mathbf x})\phi_i({\mathbf y})$ where
$\{(\lambda_i,\phi_i)\}$ are eigenvalue-eigenvector pairs of the corresponding
integral operator. This infinite representation is known to converge uniformly
to the kernel $K$. We estimate the speed of this convergence in terms of the
decay rate of eigenvalues and demonstrate that for $2m$ times differentiable
kernels the first $N$ terms of the series approximate $K$ as
$\mathcal{O}\big((\sum_{i=N+1}^\infty\lambda_i)^{\frac{m}{m+n}}\big)$ or
$\mathcal{O}\big((\sum_{i=N+1}^\infty\lambda^2_i)^{\frac{m}{2m+n}}\big)$.
Finally, we demonstrate some applications of our results to a spectral
charaterization of integral operators with continuous roots and other powers.
