Notes: `f(x)=1/(1+x)^4` Use the binomial series to find the Maclaurin series for the function.

Monday, 31 July 2017

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

Recall binomial series that is convergent when $\displaystyle{\left|{x}\right|}\lt{1}$ follows:

$\displaystyle{{\left({1}+{x}\right)}}^{{k}}={\sum_{{{n}={0}}}^{\infty}}_\frac{{{k}{\left({k}-{1}\right)}{\left({k}-{2}\right)}\ldots{\left({k}-{n}+{1}\right)}}}{{{n}!}}$

or $\displaystyle{{\left({1}+{x}\right)}}^{{k}}={1}+{k}{x}+\frac{{{k}{\left({k}-{1}\right)}}}{{{2}!}}{{x}}^{{2}}+\frac{{{k}{\left({k}-{1}\right)}{\left({k}-{2}\right)}}}{{{3}!}}{{x}}^{{3}}+\frac{{{k}{\left({k}-{1}\right)}{\left({k}-{2}\right)}{\left({k}-{3}\right)}}}{{{4}!}}{{x}}^{{4}}-$ ...

For the given function $\displaystyle{f{{\left({x}\right)}}}=\frac{{1}}{{{\left({1}+{x}\right)}}^{{4}}}$ , we may apply Law of Exponents: $\displaystyle\frac{{1}}{{{x}}^{{n}}}={{x}}^{{-{n}}}$ to rewrite it as:

$\displaystyle{f{{\left({x}\right)}}}={{\left({1}+{x}\right)}}^{{-{4}}}$

This now resembles $\displaystyle{{\left({1}+{x}\right)}}^{{k}}$ for binomial series.

By comparing " $\displaystyle{{\left({1}+{x}\right)}}^{{k}}$ " with " $\displaystyle{{\left({1}+{x}\right)}}^{{-{4}}}$ ", we have the corresponding values:

$\displaystyle{x}={x}$ and $\displaystyle{k}=-{4}$ .

Plug-in the values on...

Recall binomial series that is convergent when $\displaystyle{\left|{x}\right|}\lt{1}$ follows:

$\displaystyle{{\left({1}+{x}\right)}}^{{k}}={\sum_{{{n}={0}}}^{\infty}}_\frac{{{k}{\left({k}-{1}\right)}{\left({k}-{2}\right)}\ldots{\left({k}-{n}+{1}\right)}}}{{{n}!}}$

$\displaystyle{f{{\left({x}\right)}}}={{\left({1}+{x}\right)}}^{{-{4}}}$

This now resembles $\displaystyle{{\left({1}+{x}\right)}}^{{k}}$ for binomial series.

By comparing " $\displaystyle{{\left({1}+{x}\right)}}^{{k}}$ " with " $\displaystyle{{\left({1}+{x}\right)}}^{{-{4}}}$ ", we have the corresponding values:

$\displaystyle{x}={x}$ and $\displaystyle{k}=-{4}$ .

Plug-in the values on the formula for binomial series, we get:

$\displaystyle{{\left({1}+{x}\right)}}^{{-{4}}}={\sum_{{{n}={0}}}^{\infty}}\frac{{{\left(-{4}\right)}{\left(-{4}-{1}\right)}{\left(-{4}-{2}\right)}\ldots{\left(-{4}-{n}+{1}\right)}}}{{{n}!}}{{x}}^{{n}}$

$\displaystyle={1}+{\left(-{4}\right)}{x}+\frac{{{\left(-{4}\right)}{\left(-{4}-{1}\right)}}}{{{2}!}}{{x}}^{{2}}+\frac{{{\left(-{4}\right)}{\left(-{4}-{1}\right)}{\left(-{4}-{2}\right)}}}{{{3}!}}{{x}}^{{3}}+\frac{{{\left(-{4}\right)}{\left(-{4}-{1}\right)}{\left(-{4}-{2}\right)}{\left(-{4}-{3}\right)}}}{{{4}!}}{{x}}^{{4}}-$ ...

$\displaystyle={1}+{\left(-{4}\right)}{x}+\frac{{{\left(-{4}\right)}{\left(-{5}\right)}}}{{{2}!}}{{x}}^{{2}}+\frac{{{\left(-{4}\right)}{\left(-{5}\right)}{\left(-{6}\right)}}}{{{3}!}}{{x}}^{{3}}+\frac{{{\left(-{4}\right)}{\left(-{5}\right)}{\left(-{6}\right)}{\left(-{7}\right)}}}{{{4}!}}{{x}}^{{4}}-$ ...

$\displaystyle={1}-{4}{x}+\frac{{20}}{{{2}!}}{{x}}^{{2}}-\frac{{120}}{{{3}!}}{{x}}^{{3}}+\frac{{840}}{{{4}!}}{{x}}^{{4}}-$ ...

$\displaystyle={1}-{4}{x}+{10}{{x}}^{{2}}-{20}{{x}}^{{3}}+{35}{{x}}^{{4}}-$ ...

Therefore, the Maclaurin series for the function $\displaystyle{f{{\left({x}\right)}}}=\frac{{1}}{{{\left({1}+{x}\right)}}^{{4}}}$ can be expressed as:

$\displaystyle\frac{{1}}{{{\left({1}+{x}\right)}}^{{4}}}={1}-{4}{x}+{10}{{x}}^{{2}}-{20}{{x}}^{{3}}+{35}{{x}}^{{4}}-$ ...

Notes

Monday, 31 July 2017

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

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How are race, gender, and class addressed in Oliver Optic's Rich and Humble?

Monday, 31 July 2017

Use the binomial series to find the Maclaurin series for the function.

No comments:

Post a Comment

How are race, gender, and class addressed in Oliver Optic&#39;s Rich and Humble?

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

How are race, gender, and class addressed in Oliver Optic's Rich and Humble?