Get the answer to "Use the binomial series to find the Maclaurin..."

Rivka Thorpe

Answered question

2020-11-20

Use the binomial series to find the Maclaurin series for the function.
$f (x) = \frac{1}{(1 + x)^{4}}$

pierretteA

Skilled2020-11-21Added 102 answers

We have to find Maclaurin series of the Function

f (x) = \frac{1}{(1 + x)^{4}} = (1 + x)^{- 4}

by using binomial series.
We know the binomial series

(1 + x)^{α} = \sum_{n = 0}^{\infty} (\begin{matrix} α \\ n \end{matrix}) x^{n}

Where

(\begin{matrix} α \\ n \end{matrix}) = \frac{α (α - 1) (α - 2) . . . (α - n + 1)}{n!}

We have given function

f (x) = \frac{1}{(1 + x)^{4}} = (1 + x)^{- 4}

By binomial series

(1 + x)^{- 4} = \sum_{n = 0}^{\infty} (\begin{matrix} - 4 \\ n \end{matrix}) x^{n}

= \sum_{n = 0}^{\infty} \frac{(- 4) (- 4 - 1) (- 4 - 2) . . . (- 4 - n + 1)}{n!} x^{n}

= \sum_{n = 0}^{\infty} \frac{(- 4) (- 4 - 1) (- 4 - 2) . . . (- 4 - (n - 1))}{n!} x^{n}

= \sum_{n = 0}^{\infty} \frac{(- 1)^{n} 4.5 .6 .7 . . . (4 + (n - 1))}{n!} x^{n}

Therefore

f (x) = \sum_{n = 0}^{\infty} \frac{(- 1)^{n} 4.5 .6 .7 . . . (4 + (n - 1))}{n!} x^{n}

Jeffrey Jordon

Expert2021-12-27Added 2605 answers

Answer is given below (on video)

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