Write down the first four terms of the Taylor Series expansion for the expression f(x) = ...

Question:

Write down the first four terms of the Taylor Series expansion for the expression

{eq}f(x) = \frac {1}{(1 \pm x)}^n {/eq}

Binomial Series Representation:

{eq}\\ {/eq}

The standard Binomial series representation for the fractional negative powers will be used here in order to get the expression of power series for the given function {eq}\; \dfrac {1}{(1 \pm x)^{n}} \; {/eq}. First of all, we will form the series for the function {eq}\; \dfrac {1}{(1 + x)^{n}} \; {/eq} then form the series for the function {eq}\; \dfrac {1}{(1 - x)^{n}} \; {/eq} then combine them in order to get the final expression.

{eq}\displaystyle (1 + a)^{k} = 1 + ka + \dfrac {k (k-1)}{2!} \; a^{2} + \dfrac {k (k-1) (k-2)}{3!} \; a^{3} + \cdots {/eq}

The interval of convergence is given as: {eq}\; \; \Longrightarrow |a| < 1 {/eq}

{eq}\\ {/eq}

{eq}\displaystyle f(x) = \dfrac {1}{(1 - x)^{n}} {/eq}

{eq}\displaystyle (1 - x)^{-n} = 1 + nx + \dfrac {(n) \; (n+1)}{2!} \; x^{2} + \dfrac {(n) \; (n+1) \; (n +2)}{3!} \; x^{3} + \cdots {/eq}

The interval of convergence is given as: {eq}\; \; \Longrightarrow |x| < 1 {/eq}

{eq}\displaystyle (1 + x)^{-n} = 1 - nx + \dfrac {n (n +1)}{2!} \; x^{2} - \dfrac {n (n + 1) (n + 2)}{3!} \; x^{3} + \cdots {/eq}

Finally, the power series representation is given as:

{eq}\displaystyle \Longrightarrow \boxed {f(x) = \dfrac {1}{(1 \pm x)^{n}} = 1 \pm nx + \dfrac {n (n +1)}{2!} \; x^{2} \pm \dfrac {n (n + 1) (n +2)}{3!} \; x^{3} + \cdots} {/eq}

The interval of convergence is given as: {eq}\; \; \Longrightarrow \boxed { -1 < x < 1} {/eq}

How to Use the Binomial Theorem to Expand a Binomial

from Algebra II Textbook

Chapter 21 / Lesson 16
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