Picard's Method of Successive Approximations

Create a diagram illustrating Picard's method of successive approximations. First, draw a slope field on the coordinate plane for the differential equation y' = 2x(y+1), shown as short line segments whose slopes vary with x and y. Then, overlay a curve representing the k-th approximation, given by the sum from n=1 to k of t^(2n)/n!. This curve starts from the initial point and approaches the exact solution as k increases. The diagram should clearly show the relationship between the slope field and the approximation curve, with the curve superimposed on the field to demonstrate the iterative approximation process.