Up to degree 2_1 write the Taylor series for the f(x) = Cx + D^-4 centered at zero.

Question:

Up to degree 2_1 write the Taylor series for the f(x) = (Cx + D)^{-4} centered at zero.

Binomial Series Representation:

{eq}\\ {/eq}

To represent the given function as a Taylor series representation, we will use the standard Binomial series expansion for the fractional negative powers. First of all, we will convert the given function into the required form then we will proceed with the series expansion.

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

The above series expansion holds only when: {eq}\; \; \Longrightarrow |a| < 1 {/eq}

Answer and Explanation:

{eq}\\ {/eq}

{eq}f(x) = \dfrac {1}{(Cx + D)^{4}} {/eq}

{eq}\displaystyle \Longrightarrow f(x) = \dfrac {\biggr( \dfrac {1}{D^{4}} \biggr)}{\Biggr[ 1 + \biggr( \dfrac {Cx}{D} \biggr) \Biggr]^{4}} {/eq}

We know the standard Binomial series expansion for the fractional negative powers:

{eq}(1 + a)^{-4} = 1 - 4a + \dfrac {(-4) \; (-4 - 1)}{2!} \; a^{2} + \dfrac {(-4) \; (-4-1) \; (-4-2)}{3!} \; a^{3} + \cdots {/eq}

{eq}\displaystyle \Longrightarrow (1 + a)^{-4} = 1 - 4a + 10a^{2} - 20a^{3} + \cdots {/eq}

The above series expansion holds good only when: {eq}\; \; \Longrightarrow |a| < 1 {/eq}

Now replace the value of {eq}\; a = \dfrac {Cx}{D} \; {/eq} in the above expression:

{eq}\displaystyle \Biggr[1 + \biggr( \dfrac {Cx}{D} \biggr) \Biggr]^{-4} = 1 - 4 \; \biggr( \dfrac {Cx}{D} \biggr) + 10 \; \biggr( \dfrac {Cx}{D} \biggr)^{2} - 20 \; \biggr( \dfrac {Cx}{D} \biggr)^{3} + \cdots {/eq}

{eq}\displaystyle \Longrightarrow \boxed {f(x) = \dfrac {1}{(Cx + D)^{-4}} = \biggr( \dfrac {1}{D^{4}} \biggr) \; \Biggr[ 1 - 4 \; \biggr( \dfrac {Cx}{D} \biggr) + 10 \; \biggr( \dfrac {Cx}{D} \biggr)^{2} - 20 \; \biggr( \dfrac {Cx}{D} \biggr)^{3} + \cdots \Biggr] } {/eq}

The above series representation is valid only when: {eq}\; \; |\biggr( \dfrac {Cx}{D} \biggr)| < 1 \; \; \; \Longrightarrow \; \; \boxed {|x| < \biggr( \dfrac {D}{C} \biggr)} {/eq}


Learn more about this topic:

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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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