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

Use the Maclaurin series for

#x^3/(2-x^3) = sum_(n=0)^oo 2^(-n-1) x^(3n+3)#

with radius of convergence

Then we find:

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To find a power series representation for ( \frac{x^3}{2 - x^3} ), you can use partial fraction decomposition and then express each term as a geometric series. The function can be written as ( \frac{1}{2} \cdot \frac{x^3}{1 - (x^3/2)} ). By recognizing that ( \frac{x^3}{2} ) is a common ratio, you can rewrite the expression as a geometric series. The resulting power series representation is ( \sum_{n=0}^{\infty} \frac{x^{3n+3}}{2^{n+1}} ).

The radius of convergence ( R ) can be found using the ratio test. Applying the ratio test to the series, you'll find that ( R = 2^{1/3} ).

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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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