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

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asked 2020-11-20

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

## Answers (1)

2020-11-21
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
$$(1+x)^\alpha=\sum_{n=0}^\infty\left(\begin{array}{c}\alpha\\ n\end{array}\right)x^n$$
Where $$\left(\begin{array}{c}\alpha\\ n\end{array}\right)=\frac{\alpha(\alpha-1)(\alpha-2)...(\alpha-n+1)}{n!}$$
We have given function
$$f(x)=\frac{1}{(1+x)^4}=(1+x)^{-4}$$
By binomial series
$$(1+x)^{-4}=\sum_{n=0}^\infty\left(\begin{array}{c}-4\\ n\end{array}\right)x^n$$
$$=\sum_{n=0}^\infty\frac{(-4)(-4-1)(-4-2)...(-4-n+1)}{n!}x^n$$
$$=\sum_{n=0}^\infty\frac{(-4)(-4-1)(-4-2)...(-4-(n-1))}{n!}x^n$$
$$=\sum_{n=0}^\infty\frac{(-1)^n4.5.6.7...(4+(n-1))}{n!}x^n$$
Therefore $$f(x)=\sum_{n=0}^\infty\frac{(-1)^n4.5.6.7...(4+(n-1))}{n!}x^n$$

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