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

Answer 1
The radius of convergence of the binomial series is #1#.

Let's examine a few specifics.

This is how the binomial series appears:

#(1+x)^alpha=sum_{n=0}^infty((alpha),(n))x^n#, where
#((alpha),(n))={alpha(alpha-1)(alpha-2)cdots(alpha-n+1)}/{n!}#

Using the Ratio Test

#lim_{n to infty}|{a_{n+1}}/{a_n}|=lim_{n to infty}|{((alpha),(n+1))x^{n+1}}/{((alpha),(n))x^n}|#
#=lim_{n to infty}|{{alpha(alpha-1)(alpha-2)cdots(alpha-n+1)(alpha-n)}/{(n+1)!}x^{n+1}}/{{alpha(alpha-1)(alpha-2)cdots(alpha-n+1)}/{n!}x^n}|#

Eliminating all common elements,

#=lim_{n to infty}|{alpha-n}/{n+1}x|#
by pulling #|x|# out of the limit,
#=|x|lim_{n to infty}|{alpha-n}/{n+1}|#
by dividing the numerator and the denominator by #n#,
#=|x|lim_{n to infty}|{alpha/n-1}/{1+1/n}|=|x||{0-1}/{1+0}|=|x|<1#
Hence, the radius of convergence is #1#.
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Answer 2

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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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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