# How do you find the radius of convergence of the binomial power series?

Let's examine a few specifics.

This is how the binomial series appears:

Using the Ratio Test

Eliminating all common elements,

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The radius of convergence ( R ) of the binomial power series can be found using the ratio test. The ratio test states that for a power series ( \sum_{n=0}^{\infty} c_n(x-a)^n ), the radius of convergence is given by:

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

Where ( c_n ) are the coefficients of the series. For the binomial power series ( (1+x)^n ), the coefficients are given by the binomial theorem as ( c_n = \binom{n}{k} ), where ( k ) is the term number.

Therefore, the ratio of consecutive coefficients is:

[ \left| \frac{c_{n}}{c_{n+1}} \right| = \left| \frac{\binom{n}{k}}{\binom{n+1}{k}} \right| ]

Taking the limit as ( n \to \infty ), we can simplify this expression to find the radius of convergence.

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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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