What is the Taylor series expansion of #f(x) = 1/x^2# at a=1?

Answer 1

The Taylor series expansion, in general, is written as:

#sum_(n=0)^(oo) f^n(a)/(n!)(x-a)^n#
So, we will have to take #n# derivatives of #1/x^2#. #n = 3# is the bare minimum in my opinion if you want to see a significant chunk of a pattern, but let's just stop at #n = 4#; this derivative isn't too bad, I guess. You just get pretty large numbers past #n = 4#.
#f^0(x) = color(blue)(f(x)) = x^-2 color(blue)(= 1/x^2)#
#color(blue)(f'(x)) = -2x^-3 color(blue)(= -2/x^3)#
#color(blue)(f''(x)) = 6x^-4 color(blue)(= 6/x^4)#
#color(blue)(f'''(x)) = -24x^-5 color(blue)(= -24/x^5)#
#color(blue)(f''''(x)) = 120x^-6 = color(blue)(120/x^6)#

Now, let's plug them in. Just remember:

Generally, the sum is written out to be: #= f^0(1)/(0!)(x-1)^0 + (f'(1))/(1!)(x-1)^1 + (f''(1))/(2!)(x-1)^2 + (f'''(1))/(3!)(x-1)^3 + (f''''(1))/(4!)(x-1)^4 + ...#
Now plug in your newly-acquired derivatives: #= (1/(1^2))/(1!)(1) + (-2/1^3)/(1!)(x-1) + (6/1^4)/(2!)(x-1)^2 + (-24/1^5)/(3!)(x-1)^3 + (120/1^6)/(4!)(x-1)^4 + ...#
Simplify: #= 1 + (-2)(x-1) + (6)/(2)(x-1)^2 + (-24)/(6)(x-1)^3 + (120)/(24)(x-1)^4 + ...#
And simplify some more: #= color(blue)(1 - 2(x-1) + 3(x-1)^2 - 4(x-1)^3 + 5(x-1)^4 + ...)#
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Answer 2

The Taylor series expansion of ( f(x) = \frac{1}{x^2} ) at ( a = 1 ) is:

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

By finding the derivatives of ( f(x) ) at ( a = 1 ) and evaluating them:

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

Evaluating at ( a = 1 ):

[ f(1) = 1 ] [ f'(1) = -2 ] [ f''(1) = 6 ] [ f'''(1) = -24 ]

The Taylor series expansion becomes:

[ f(x) = 1 - 2(x-1) + 6(x-1)^2 - 24(x-1)^3 + \ldots ]

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Answer from HIX Tutor

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.

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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