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

The Taylor series can be written out as:

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# 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!) #

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To find the Taylor series expansion of (e^{-\frac{x^2}{2}}), we use the standard formula for the exponential function's Taylor series. The Taylor series expansion of (e^x) around (x = a) is given by:

[ e^x = \sum_{n=0}^{\infty} \frac{{f^{(n)}(a)}}{{n!}}(x - a)^n ]

Here, (f^{(n)}(a)) represents the (n)th derivative of (f(x)) evaluated at (x = a). For (e^{-\frac{x^2}{2}}), (a = 0).

To find the derivatives of (e^{-\frac{x^2}{2}}), use the chain rule. The first few derivatives are:

[ f(x) = e^{-\frac{x^2}{2}} ] [ f'(x) = -x e^{-\frac{x^2}{2}} ] [ f''(x) = (x^2 - 1) e^{-\frac{x^2}{2}} ] [ f'''(x) = (3x - x^3) e^{-\frac{x^2}{2}} ]

Substitute these derivatives into the Taylor series formula and evaluate them at (a = 0) to find the coefficients. The Taylor series expansion of (e^{-\frac{x^2}{2}}) around (x = 0) will be the sum of these terms.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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