The functions e^x, e^-x, e^(-x^2), erf(x) and Taylor Series

The functions e^x, e^-x, e^(-x^2), erf(x) and Taylor Series

Accurate digits are highlighted in green.  Calculations are used with a TI 84 Plus CE.

e^x = 1 + x + x^2/2! + x^3/3! + x^4/4 + … = Σ(x^n/n!, from n = 0 to ∞)

x =	e^x	10 terms	25 terms	50 terms
1	2.718281828	2.718281801	2.718281828	2.718281828
3	20.08553692	20.07966518	20.08553692	20.08553692
5	148.4131591	146.380601	148.4131591	148.4131591
9.9	19930.37044	11869.50538	19930.07221	19930.37044

e^(-x) = 1 – x + x^2/2! – x^3/3! + x^4/4 - … = Σ( (-x)^n/n!, from n = 0 to ∞)

x =	e^(-x)	10 terms	25 terms	50 terms
1	0.3678794412	0.3678794643	0.3678794412	0.3678794412
3	0.0497870684	0.0533258929	0.0497870684	0.0497870684
5	0.006737947	0.8640390763	0.0067379439	0.006737947
9.9	5.017468206E-5	1207.799663	-0.1392914019	5.017463241E-5

I think you know where I’m going.

e^(-x^2) = 1 – x^2 + x^4/2! – x^6/3! + x^8/4! = Σ( (-x)^(2*n)/n!, from n = 0 to ∞)

x =	e^(-x^2)	10 terms	25 terms	50 terms
1	0.3678794412	0.3678794643	0.3678794412	0.3678794412
3	1.234098041E-4	442.2750223	-0.0118646275	1.234194001E-4
5	1.38879439E-11	18613495.8	-2834107793	85689.40174
9.9	2.72143414E-43	2.04347238E13	-3.10254183E24	7.951057508E34

(Something really goes bonkers as x increases and n increases)

Error Function

erf(x) = 2/√π * ∫(e^(-t^2) dt, 0, x)

= 2/√π * (x – x^3/3 + x^5/(5*2!) – x^7/(7*3!) + x^9/(9*4!) - ...)

= 2/√π * Σ( (-x^(2n+1)/((2n+1)*n!) from n = 0 to ∞ )

x =	erf(x)	10 terms	25 terms	50 terms
1	0.8427007929	0.8427007941	0.8427007929	0.8427007929
3	0.9999779095	68.58627744	0.9992050426	0.9999779095
5	1	4853382.901	-3070260210.4	4724.331354
9.9	1	1.076461715E13	-6.7395908E23	*overflows during calculation* (Result: 8.73442E33 from WolframAlpha) (erf(x) is practically 1 for x > 3)

Note: 9.9^(2*50+1) ≈ 3.623E100

Thoughts:

*  Taylor series are great when x is near its center point.  In the all the cases above, the center point is x = 0. 

*  The more simple the expression, the better range of accuracy with less terms. 

*  Before you recommend a Taylor Series to approximate f(x), check the accuracy and the range.  A cautionary tale. 

Eddie

This blog is property of Edward Shore, 2016.