Another Gaussian approximation

The function (1 + cos(x))/2 gives a fair approximation to the Gaussian density exp(−x²) You can make the approximation much better by raising it to a power. The function ((1 + cos(x))/2)4 gives a good lower bound and ((1 + cos(x))/2)3.5597 gives a good upper bound. More on that here. There are other ways of improving the cosine approximation to the Gaussian. Yesterday I came across one I hadn’t seen before, adding a sin(x) term to x. (1 + cos(sin(x) + x))/2 This function matches the first few terms of the power series for exp(−x²) and has an error on the order of x6/240. You can’t see the difference between the two functions in a plot for −4 ≤ x ≤ 4. There’s some tension between the previous two statements. If the error in on the order of x6/240 then we’d expect the error to be huge at x = 4. We have an alternating series, so the truncation error should be roughly equal to the first term after the truncation, right? No, the alternating series theorem doesn’t apply because the absolute…