# How do you find a power series representation for # (x/(2-x))^3# and what is the radius of convergence?

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To find the power series representation of ((\frac{x}{2-x})^3), we can use the geometric series expansion:

[ \frac{1}{1 - (-\frac{x}{2})} = \sum_{n=0}^\infty (-1)^n (\frac{x}{2})^n ]

Now, we raise both sides to the power of 3:

[ (\frac{1}{1 - (-\frac{x}{2})})^3 = \sum_{n=0}^\infty (-1)^n (\frac{x}{2})^n ]

After expanding and simplifying, we get:

[ (\frac{1}{1 + \frac{x}{2}})^3 = \sum_{n=0}^\infty \binom{n+2}{2} (\frac{x}{2})^n ]

So, the power series representation for ((\frac{x}{2-x})^3) is:

[ \sum_{n=0}^\infty \binom{n+2}{2} (\frac{x}{2})^n ]

The radius of convergence can be found using the ratio test:

[ R = \lim_{n \to \infty} \frac{a_{n}}{a_{n+1}} ]

where (a_{n}) is the coefficient of (x^{n}) term in the series. In this case, (a_{n} = \binom{n+2}{2}). The limit can be computed to find the radius of convergence.

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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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