Final answer:
To find the Taylor series of f(x) = 8/x centered at a = -2, one must compute the derivatives of the function at x = -2 and apply them in the Taylor series formula. The first few derivatives evaluated at a = -2 give us the beginning of the series, which consists of: ... -4 + 2(x + 2) - (x + 2)² + ...
Step-by-step explanation:
Finding the Taylor Series for f(x) = 8/x Centered at a = -2
To find the Taylor series of the function f(x) = 8/x centered at a = -2, we need to calculate the derivatives of f(x) at x = -2 and then form the series using the Taylor series formula. The Taylor series for a function f(x) around a point a is given by:
f(a) + f'(a)(x - a) + ⅗ f''(a)(x - a)² + ⅗ f'''(a)(x - a)³ + ...
Let's compute the first few derivatives of f(x):
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- f(x) = 8/x
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- f'(x) = -8/x²
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- f''(x) = 16/x³
Now, evaluate the derivatives at a = -2:
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- f(-2) = -4
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- f'(-2) = 2
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- f''(-2) = -2
Using these values in the Taylor series formula, the series for f(x) centered at a = -2 is:
... -4 + 2(x + 2) - (x + 2)² + ...
This is the general approach for finding the Taylor series. To write down more terms, you would need to continue the process of differentiation, evaluate at a = -2, and add the terms to the series.