# Find the Taylor expansion #\color(red)\bb\text(formula)#... for #f(x)=1/x^2# given #a=4#?

##
Please check my work (it has to be done using the colored parts, sorry):

#f'=(-2)x^-3#

#f''=(-2)(-3)x^-4#

#f'''=(-2)(-3)(-4)x^-5#

#f'(4)=(-2)*4^-3#

#f''(4)=(-2)(-3)*4^-4#

#f'''(4)=(-2)(-3)(-4)*4^-5#

#\color(green)(f^n(4))=(-1)^n(n-1)!4^-n=\color(olive)(((-1)^n(n-1)!)/4^n)#

#\color(red)(C_n=f^n(a)*1/(n!))=((-1)^n(n-1)!)/4^n*1/(n!)=((-1)^n(n-1))/4^n#

#\rarr\color(red)(f(x)=\sum_(n=0)^\inftyC_n(x-a)^n)=\sum_(n=0)^\infty((-1)^n(n-1))/4^n*(x-2)^n#

(Can I simplify this further?)

Please check my work (it has to be done using the colored parts, sorry):

(Can I simplify this further?)

Thus,

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The Taylor expansion formula for a function ( f(x) ) centered at ( a ) is:

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

For ( f(x) = \frac{1}{x^2} ) centered at ( a = 4 ), we need to find the derivatives of ( f(x) ) and evaluate them at ( x = 4 ). Then, we substitute these values into the Taylor series formula.

First, let's find the derivatives of ( f(x) ):

[ f(x) = \frac{1}{x^2} ] [ f'(x) = -\frac{2}{x^3} ] [ f''(x) = \frac{6}{x^4} ] [ f'''(x) = -\frac{24}{x^5} ]

Now, let's evaluate these derivatives at ( x = 4 ):

[ f(4) = \frac{1}{4^2} = \frac{1}{16} ] [ f'(4) = -\frac{2}{4^3} = -\frac{1}{8} ] [ f''(4) = \frac{6}{4^4} = \frac{3}{32} ] [ f'''(4) = -\frac{24}{4^5} = -\frac{3}{16} ]

Now, we substitute these values into the Taylor series formula:

[ f(x) = f(4) + f'(4)(x - 4) + \frac{f''(4)}{2!}(x - 4)^2 + \frac{f'''(4)}{3!}(x - 4)^3 + \ldots ]

[ = \frac{1}{16} - \frac{1}{8}(x - 4) + \frac{3}{64}(x - 4)^2 - \frac{1}{16}(x - 4)^3 + \ldots ]

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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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