Question

How to do taylor series expansion of `e^(-x^2"/2")`?

Answer 1

When you construct a Taylor series, you're going to have to take some `n`th order derivatives. To get a substantial way into the series, let's try going to `n = 4`.

The Taylor series can be written out as:

`sum_(n=0)^N (f^((n))(a))/(n!)(x-a)^n`

You didn't specify what `a` was, but I will just assume a general case of `a = a`.

`color(green)(f^((0))(x)) = f(x) = color(green)(e^(-x^2"/2"))`

`color(green)(f'(x)) = e^(-x^2"/2") * -x = color(green)(-xe^(-x^2"/2"))`

`color(green)(f''(x)) = (-x)[-xe^(-x^2"/2")] + e^(-x^2"/2")(-1)`
`= x^2e^(-x^2"/2") - e^(-x^2"/2")`
`= color(green)((x^2 - 1)e^(-x^2"/2"))`

`color(green)(f'''(x)) = (x^2 - 1)(-xe^(-x^2"/2")) + e^(-x^2"/2")(2x)`
`= (-x^3 + x)(e^(-x^2"/2")) + 2xe^(-x^2"/2")`
`= color(green)((-x^3 + 3x)e^(-x^2"/2"))`

`color(green)(f''''(x)) = (-x^3 + 3x)(-xe^(-x^2"/2")) + e^(-x^2"/2")(-3x^2 + 3)`
`= (x^4 - 3x^2)e^(-x^2"/2") - 3x^2e^(-x^2"/2") + 3e^(-x^2"/2")`
`= color(green)((x^4 - 6x^2 + 3)e^(-x^2"/2"))`

Then we can write this out, using `x->a` and `n = 4` terms:

`sum_(n=0)^4 (f^((n))(a))/(n!)(x-a)^n`

`= (f^((0))(a))/(0!)(x-a)^0 + (f'(a))/(1!)(x-a)^1 + (f''(a))/(2!)(x-a)^2 + (f'''(a))/(3!)(x-a)^3 + (f''''(a))/(4!)(x-a)^4 + ...`

`= e^(-a^2"/2") + (-ae^(-a^2"/2"))(x-a) + ((a^2 - 1)e^(-a^2"/2"))/(2)(x-a)^2 + ((-a^3 + 3a)e^(-a^2"/2"))/(6)(x-a)^3 + ((a^4 - 6a^2 + 3)e^(-a^2"/2"))/(4!)(x-a)^4 + ...`

`= color(blue)(e^(-a^2"/2") - ae^(-a^2"/2")(x-a) + ((a^2 - 1)e^(-a^2"/2"))/(2)(x-a)^2 + ((-a^3 + 3a)e^(-a^2"/2"))/(6)(x-a)^3 + ((a^4 - 6a^2 + 3)e^(-a^2"/2"))/(4!)(x-a)^4 + ...)`

If we have `a = 0`, this simplifies significantly:

`= color(blue)(1 - x^2/2 + x^4/8 - x^6/48 + ...)`

Answer 2

`e^(-x^2/2)= 1 - x^2/2 + x^4/8 - x^6/48 + x^8/384 - x^10/3840 + ...`

or,

`e^(-x^2/2) = sum_(n=0)^oo (-1)^(n)x^(2n)/(2^n n!)`

Explanation:

Assuming a TS pivoted about `x=0` (ie a MacLaurin Series) we start with well known series:

`e^x= 1 + x + (x^2)/(2!) + (x^3)/(3!) + (x^4)/(4!) + (x^5)/(5!) + ...`

Replacing "`x`"with `-x^2/2` we get:

`e^(-x^2/2)= 1 + (-x^2/2) + ((-x^2/2)^2)/(2!) + ((-x^2/2)^3)/(3!) + ((-x^2/2)^4)/(4!) + ((-x^2/2)^5)/(5!) + ...`

`\ \ \ \ \ \ = 1 - x^2/2 + (x^4/2^2)/(2) - (x^6/2^3)/(6) + (x^8/2^4)/(24) - (x^10/2^5)/(120) + ...`

`\ \ \ \ \ \ = 1 - x^2/2 + (x^4/4)/(2) - (x^6/8)/(6) + (x^8/16)/(24) - (x^10/32)/(120) + ...`

`\ \ \ \ \ \ = 1 - x^2/2 + x^4/8 - x^6/48 + x^8/384 - x^10/3840 + ...`

And we note that the `n^(th)` term is given by:

`u_n = (-1)^(n)x^(2n)/(2^n n!)`