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

This serie is convergent to

#{ (A+B=1), (2A+B=0) :}#

or

Finally

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To find the power series representation for ( f(x) = \frac{x}{x^2 - 3x + 2} ), we first factor the denominator to ( (x - 1)(x - 2) ). Then, we express ( f(x) ) as a partial fraction:

[ f(x) = \frac{x}{(x - 1)(x - 2)} = \frac{A}{x - 1} + \frac{B}{x - 2} ]

Solving for ( A ) and ( B ), we get ( A = -2 ) and ( B = 3 ). Now, we can rewrite ( f(x) ) as:

[ f(x) = \frac{-2}{x - 1} + \frac{3}{x - 2} ]

Next, we express each term as a geometric series:

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

Combining these, we have:

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

Finally, we obtain the power series representation for ( f(x) ):

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

The radius of convergence of this power series is the distance from the center of the series to the nearest singularity. In this case, the singularities occur at ( x = 1 ) and ( x = 2 ). Therefore, the radius of convergence is the distance from the center, which is ( x = 1.5 ), to the nearest singularity, which is ( x = 1 ) or ( x = 2 ). So, the radius of convergence is ( 0.5 ).

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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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