Question

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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