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Given that (1 + x)^k = 1+ kx + \frac{k(k - 1)}{2!}x^2 + \frac{k(k - 1)(k - 2)}{3!}x^3 + ... ,...

Question:

Given that {eq}\displaystyle (1 + x)^k = 1+ kx + \frac{k(k - 1)}{2!}x^2 + \frac{k(k - 1)(k - 2)}{3!}x^3 + ... , {/eq} find the first four terms of the binomial series for {eq}\displaystyle f(x) = \frac{x}{(1 + x^2)^5} {/eq}.

Binomial Series Representation:

{eq}\\ {/eq}

The Binomial series representation for the negative powers can be used here in order to get the expression of series representation along with the interval of convergence for the given function. First of all, we will determine the power series for the related function with the help of the method of substitution then we will go for the whole complete function.

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

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

Answer and Explanation:

{eq}\\ {/eq}

{eq}\displaystyle f(x) = \dfrac {x}{(1 + x^{2})^{5}} {/eq}

We know the power series representation of the function {eq}\; (1 + x)^{k} \; {/eq}:

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

The above series representation is valid only when: {eq}\; \; \Longrightarrow \; |x| \; < \; 1 {/eq}

Now put the value of {eq}\; k = - 5 \; {/eq} in the above expression:

{eq}\displaystyle (1 + x)^{-5} = 1 - 5x + \dfrac {(-5) \; (-5 -1)}{2!} \; x^{2} + \dfrac {(-5) \; (-5 -1) \; (-5 -2)}{3!} \; x^{3} + \cdots {/eq}

{eq}\displaystyle (1 + x)^{-5} = 1 - 5x + 15x^{2} - 35x^{3} + \cdots {/eq}

Now replace the value of {eq}\; x \; {/eq} with {eq}\; x^{2} \; {/eq} in the above expression:

{eq}\displaystyle (1 + x^{2})^{-5} = 1 - 5x^{2} + 15x^{4} - 35x^{6} + \cdots {/eq}

The interval of convergence is given as: {eq}\; \; |x^{2}| \; < \; 1 \; \; \; \Longrightarrow \; |x| < 1 {/eq}

Finally, we have the power series representation of the function {eq}\; f(x) \; {/eq}:

{eq}\displaystyle \Longrightarrow \boxed {f(x) = \dfrac {x}{(1 + x^{2})^{5}} = x - 5x^{3} + 15x^{5} - 35x^{7} + \cdots } {/eq}

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