What is the Taylor series expansion of #f(x) = 1/x^2# at a=1?
The Taylor series expansion, in general, is written as:
Now, let's plug them in. Just remember:
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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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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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