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

Answer 1

#(x/(2-x))^3 equiv sum_(k=2)^oo(k(k-1))/2^(k+2)x^(k+1)# which is convergent for #abs(x) < 2#

We have #(x/(2-x))^3=(x/2)^3(1/(1-x/2))^3#
Calling #y = x/2# we have
#(x/(2-x))^3 equiv y^3(1/(1-y))^3#

but

#d^2/dy^2(1/(1-y))=2(1/(1-y))^3#
Now, for #abs y < 1# we have
#1/(1-y)=sum_(k=0)^ooy^k# then
#y^3(1/(1-y))^3 = y^3/2d^2/dy^2 sum_(k=0)^ooy^k=y^3/2 sum_(k=2)^ook(k-1)y^(k-2)# or
#y^3(1/(1-y))^3=1/2sum_(k=2)^ook(k-1)y^(k+1)#
but #y=x/2# so finally
#(x/(2-x))^3 equiv sum_(k=2)^oo(k(k-1))/2^(k+2)x^(k+1)# which is convergent for #abs(x/2) < 1# or #abs(x) < 2#
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Answer 2

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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Answer from HIX Tutor

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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