# How do you find a power series representation for #f(x)= 1/(1+4x^2)# and what is the radius of convergence?

Consider the power series:

Then:

So

That is:

In general

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To find a power series representation for ( f(x) = \frac{1}{1 + 4x^2} ), we can use the geometric series expansion ( \frac{1}{1 - u} = \sum_{n=0}^{\infty} u^n ) for ( |u| < 1 ).

Substitute ( u = -4x^2 ) into the geometric series expansion, yielding ( \frac{1}{1 + 4x^2} = \sum_{n=0}^{\infty} (-4x^2)^n ).

This simplifies to ( \sum_{n=0}^{\infty} (-1)^n \cdot 4^n \cdot x^{2n} ).

The radius of convergence ( R ) for this power series can be determined using the ratio test, which states that ( R = \lim_{n \to \infty} \left| \frac{a_{n}}{a_{n+1}} \right| ), where ( a_{n} ) is the coefficient of ( x^n ) in the power series.

For our series, ( a_{n} = (-1)^n \cdot 4^n ), so ( \lim_{n \to \infty} \left| \frac{a_{n}}{a_{n+1}} \right| = \lim_{n \to \infty} \left| \frac{(-1)^n \cdot 4^n}{(-1)^{n+1} \cdot 4^{n+1}} \right| = \lim_{n \to \infty} \left| \frac{1}{4} \right| = \frac{1}{4} ).

Thus, the radius of convergence ( R ) is ( \frac{1}{4} ).

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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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