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 ∞)
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x =
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e^x
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10 terms
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25 terms
|
50 terms
|
|
1
|
2.718281828
|
2.718281801
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2.718281828
|
2.718281828
|
|
3
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20.08553692
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20.07966518
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20.08553692
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20.08553692
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5
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148.4131591
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146.380601
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148.4131591
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148.4131591
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9.9
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19930.37044
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11869.50538
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19930.07221
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19930.37044
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e^(-x) = 1 – x + x^2/2! –
x^3/3! + x^4/4 - … = Σ( (-x)^n/n!, from n = 0 to ∞)
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x =
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e^(-x)
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10 terms
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25 terms
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50 terms
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1
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0.3678794412
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0.3678794643
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0.3678794412
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0.3678794412
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3
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0.0497870684
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0.0533258929
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0.0497870684
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0.0497870684
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5
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0.006737947
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0.8640390763
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0.0067379439
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0.006737947
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9.9
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5.017468206E-5
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1207.799663
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-0.1392914019
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5.017463241E-5
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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 ∞)
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x =
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e^(-x^2)
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10 terms
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25 terms
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50 terms
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|
1
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0.3678794412
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0.3678794643
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0.3678794412
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0.3678794412
|
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3
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1.234098041E-4
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442.2750223
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-0.0118646275
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1.234194001E-4
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5
|
1.38879439E-11
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18613495.8
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-2834107793
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85689.40174
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9.9
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2.72143414E-43
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2.04347238E13
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-3.10254183E24
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7.951057508E34
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(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 ∞ )
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x =
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erf(x)
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10 terms
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25 terms
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50 terms
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1
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0.8427007929
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0.8427007941
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0.8427007929
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0.8427007929
|
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3
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0.9999779095
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68.58627744
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0.9992050426
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0.9999779095
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5
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1
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4853382.901
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-3070260210.4
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4724.331354
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9.9
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1
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1.076461715E13
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-6.7395908E23
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*overflows during
calculation*
(Result: 8.73442E33 from
WolframAlpha)
(erf(x) is practically 1
for x > 3)
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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.
