In Part 1 and Part 2, the series approximation for the integrated blackbody spectrum was

(3.1)

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

(3.2)

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.