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Find the binomial series for the function f(x)=\frac{1}{\sqrt [3]{1 + x^{4}}} and simplify the...

Question:

Find the binomial series for the function {eq}\displaystyle f(x) = \frac{1}{\sqrt [3]{1 + x^{4}}} {/eq} and simplify the coefficients in sum notation.

Basics of Binomial Series Expansion:

If {eq}k {/eq} is any number and {eq}\left| x \right| < 1 {/eq} then Binomial series expansion is given as:

{eq}{(1 + x)^k} = \sum\limits_{n = 0}^\infty {\left( {\begin{array}{*{20}{c}} k \\ n \\ \end{array}} \right){x^n}} = 1 + kx + \frac{{k(k - 1)}}{{2!}}{x^2} + \frac{{k(k - 1)(k - 2)}}{{3!}}{x^3} + ..... {/eq}

Answer and Explanation:

Here in this case, the given function is represented and expanded as:

{eq}\eqalign{ f(x)& = \frac{1}{{\root 3 \of {1 + {x^4}} }} \cr & = {\left(...

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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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