# How do you find a power series converging to #f(x)=x/(1+x)^4# and determine the radius of convergence?

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To find a power series converging to ( f(x) = \frac{x}{(1+x)^4} ) and determine the radius of convergence, you can use the method of differentiating or integrating a known power series. Since ( \frac{1}{1+x} ) has a known power series expansion, we can start from there. We know that ( \frac{1}{(1+x)^4} ) is the fourth derivative of ( \frac{1}{1+x} ). Hence, by differentiating the known power series for ( \frac{1}{1+x} ) four times, and then multiplying by ( x ), we can obtain the power series for ( f(x) ).

After obtaining the power series representation of ( f(x) ), you can then determine the radius of convergence using the ratio test or the root test. These tests will provide the interval of convergence, and the radius of convergence will be the distance from the center of the interval to either endpoint.

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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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