In Part 1 and Part 2, the series approximation for the integrated blackbody spectrum was
Having two separate formulas is awkward in practice. Combining the two requires a function that differentiates between small and large v, going to one for small v and to zero for large v. This type of function is a variation of the Sigmoid function. After trying various functions, the first form that worked okay was e-v2. After applying this function to both sides of equation 1.6 from Part 1, the result is
A plot of the first 5 partial sums is shown in Figure 1.
The new fit clearly works well for large and small v. For v near 1, the fit appears to converge slowly or not at all. The steep, downward spikes are places where the fit crosses the true function, so the error is zero.