How do you find the maclaurin series expansion of #f(x) = 1/((1+x)^2)#?
The answer is
The series Maclaurin is
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To find the Maclaurin series expansion of ( f(x) = \frac{1}{{(1+x)^2}} ), you can start by expressing ( f(x) ) as a power series by using the geometric series formula. Then differentiate ( f(x) ) and evaluate it at ( x = 0 ) to find the coefficients of the Maclaurin series expansion.

Start with the geometric series formula: [ \frac{1}{{1  u}} = 1 + u + u^2 + u^3 + \ldots ] where ( u = x ).

Differentiate ( f(x) = \frac{1}{{(1+x)^2}} ) to find ( f'(x) ).
[ f'(x) = 2(1+x)^{3} ]

Evaluate ( f'(x) ) at ( x = 0 ) to find the coefficient of the linear term in the expansion. [ f'(0) = 2 ]

Use this coefficient in the series expansion.
[ f(x) = f(0) + f'(0)x + \frac{f''(0)x^2}{2!} + \frac{f'''(0)x^3}{3!} + \ldots ]
 Substitute the value of ( f'(0) ) into the expansion.
[ f(x) = 1  2x + \ldots ]
So, the Maclaurin series expansion of ( f(x) = \frac{1}{{(1+x)^2}} ) is ( 1  2x + \ldots ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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