# How do you find a power series representation for #f(x) = (x) / ((1-x)^2)# and what is the radius of convergence?

Recall that:

What you can do is take the derivative of both sides:

So, all we need to do is do it for the explicit series to get:

Therefore, the radius of converge is:

or you can write it as:

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To find the power series representation of ( f(x) = \frac{x}{(1-x)^2} ), we first find its derivative and integrate term by term.

The function ( f(x) = \frac{x}{(1-x)^2} ) can be expressed as ( f(x) = x(1-x)^{-2} ).

Now, let's find the derivatives:

- ( f(x) = x(1-x)^{-2} )
- ( f'(x) = (1-x)^{-2} - 2x(1-x)^{-3} )
- ( f''(x) = 2(1-x)^{-3} - 6x(1-x)^{-4} )

And so on. Each derivative introduces a higher power of ( (1-x) ) in the denominator.

Now, we evaluate each derivative at ( x = 0 ) to find the coefficients of the power series expansion:

- ( f(0) = 0 )
- ( f'(0) = 1 )
- ( f''(0) = 2 )
- ( f'''(0) = 6 )

Thus, the coefficients of the power series expansion are 0, 1, 2, 6, and so on.

The power series representation of ( f(x) ) is therefore:

[ f(x) = \sum_{n=0}^{\infty} n x^n ]

The radius of convergence for this power series is ( R = 1 ), which can be determined using the Ratio Test.

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To find the power series representation for ( f(x) = \frac{x}{(1-x)^2} ), we can first express ( \frac{1}{(1-x)^2} ) as a power series, then multiply it by ( x ).

The power series representation for ( \frac{1}{(1-x)^2} ) is ( \sum_{n=0}^{\infty} (n+1)x^n ).

Multiplying by ( x ), we get ( x \cdot \sum_{n=0}^{\infty} (n+1)x^n = \sum_{n=0}^{\infty} (n+1)x^{n+1} ).

So, the power series representation for ( f(x) ) is ( \sum_{n=0}^{\infty} (n+1)x^{n+1} ).

The radius of convergence for this power series can be found using the ratio test. Let ( a_n = n+1 ). Applying the ratio test:

[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left| \frac{n+2}{n+1} \right| = 1 ]

Since the limit is equal to 1, the radius of convergence is ( R = \frac{1}{1} = 1 ). Therefore, the radius of convergence for the power series representation of ( f(x) ) is 1.

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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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