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 , you can use partial fraction decomposition and then express each term as a geometric series. The function can be written as . By recognizing that is a common ratio, you can rewrite the expression as a geometric series. The resulting power series representation is .
The radius of convergence can be found using the ratio test. Applying the ratio test to the series, you'll find that .
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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.
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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