# How do you find the power series representation for the function #f(x)=(1+x)/(1-x)# ?

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To find the power series representation for the function ( f(x) = \frac{1+x}{1-x} ), we'll use the geometric series formula:

[ \frac{1}{1-x} = 1 + x + x^2 + x^3 + \ldots ]

First, rewrite ( f(x) ) as:

[ f(x) = \frac{1+x}{1-x} = (1+x) \cdot \frac{1}{1-x} ]

Now, substitute the geometric series formula into ( \frac{1}{1-x} ):

[ f(x) = (1+x) \cdot (1 + x + x^2 + x^3 + \ldots) ]

Next, distribute ( (1+x) ) into the series:

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

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

So, the power series representation for ( f(x) = \frac{1+x}{1-x} ) is:

[ f(x) = 1 + 2x + 2x^2 + 2x^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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