Expand the function as a binomial series. (Give the answer as the first four terms of the series)...

Question:

Expand the function as a binomial series. (Give the answer as the first four terms of the series)

{eq}f(x) = \displaystyle\frac {1}{\sqrt {1 + x^2}} {/eq}

Give the interval of the convergence of the series. (Enter answer using interval notation)

Binomial Series:

The binomial series for {eq}f(x) = (1 + x)^k {/eq} is given by the formula

{eq}(1 + x)^k = 1 + kx + \displaystyle\frac{k(k-1) x^2}{2} + \ldots + \frac{k(k-1)\ldots(k-n+1)x^n}{n!} + \ldots {/eq}

Here {eq}k {/eq} is a noninteger power, but we can also use this series for negative integer powers as well. We can then replace {eq}x {/eq} with powers of x to obtain other series as well.

Answer and Explanation:

We are given the function

{eq}f(x) = \displaystyle\frac {1}{\sqrt {1 + x^2}} = (1 + x^2)^{-1/2}. {/eq}

Therefore {eq}k =...

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How to Use the Binomial Theorem to Expand a Binomial

from Algebra II Textbook

Chapter 21 / Lesson 16
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