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Find the first four terms of the binomial series for the function shown below. (1 + x^3)^{-1/5}

Question:

Find the first four terms of the binomial series for the function shown below.

{eq}\left(1 + x^3\right)^{-\frac{1}{5}} {/eq}

Binomial Series:

In this question, we will use the Binomial theorem to calculate the first four terms of the binomial series. The binomial series is a Taylor series and it is expressed in binomial coefficient is the series.

Answer and Explanation:

Given that,

{eq}\left(1 + x^3\right)^{-\frac{1}{5}} {/eq}

The Binomial series is defined as:

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

So,

{eq}(1+x^3)^{-1/5} =1+(\dfrac{-1}{5})x^3 + \dfrac{\dfrac{-1}{5}(\dfrac{-1}{5}-1)}{2!} x^6 + \dfrac{\dfrac{-1}{5}(\dfrac{-1}{5}-1)(\dfrac{-1}{5}-2)}{3!} x^9 +.....\\ \color{blue}{(1+x^3)^{-1/5} = 1 - \dfrac{x^3}{5} + \dfrac{\dfrac{-1}{5}(\dfrac{-6}{5})}{2!}x^6 + \dfrac{\dfrac{-1}{5}(\dfrac{-6}{5})(\dfrac{-11}{5})}{3!}x^9 + .....} {/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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