# How do you find a power series representation for #f(x) = x / (1+x^2) # and what is the radius of convergence?

Write out a power series that when multiplied by

Find

So:

...if the sums converge.

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To find a power series representation for ( f(x) = \frac{x}{1+x^2} ), we can use the geometric series formula:

[ \frac{1}{1 - u} = 1 + u + u^2 + u^3 + \dots ]

First, rewrite ( f(x) ) as:

[ f(x) = x \cdot \frac{1}{1+x^2} ]

We can represent ( \frac{1}{1+x^2} ) as a geometric series by letting ( u = -x^2 ). Thus:

[ \frac{1}{1+x^2} = 1 - x^2 + x^4 - x^6 + \dots ]

Multiplying by ( x ):

[ f(x) = x(1 - x^2 + x^4 - x^6 + \dots) ]

[ f(x) = x - x^3 + x^5 - x^7 + \dots ]

This is the power series representation for ( f(x) = \frac{x}{1+x^2} ).

The radius of convergence ( R ) of a power series can be found using the ratio test formula:

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

In this case, the terms of the series alternate in sign and decrease in magnitude, suggesting that the series converges for all ( x ). Thus, the radius of convergence ( R ) is ( \infty ).

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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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