New Notes Blog: `f(x)=sqrt(1+x^2)` Use the binomial series to find the Maclaurin series for the function.

Sunday, 11 August 2013

$\displaystyle{f{{\left({x}\right)}}}=\sqrt{{{1}+{{x}}^{{2}}}}$ Use the binomial series to find the Maclaurin series for the function.

A binomial series is an example of infinite series. When we have a function of $\displaystyle{f{{\left({x}\right)}}}={{\left({1}+{x}\right)}}^{{k}}$ such that k is any number and convergent to $\displaystyle{\left|{x}\right|}\lt{1}$ , we may apply the sum of series as the value of $\displaystyle{{\left({1}+{x}\right)}}^{{k}}$ . This can be expressed in a formula:

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

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

To evaluate the given function $\displaystyle{f{{\left({x}\right)}}}=\sqrt{{{1}+{{x}}^{{2}}}}$ , we express it in term...

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

To evaluate the given function $\displaystyle{f{{\left({x}\right)}}}=\sqrt{{{1}+{{x}}^{{2}}}}$ , we express it in term of fractional exponent:

$\displaystyle{f{{\left({x}\right)}}}={{\left({1}+{{x}}^{{2}}\right)}}^{{\frac{{1}}{{2}}}}$

$\displaystyle{f{{\left({x}\right)}}}={{\left({1}+{{x}}^{{2}}\right)}}^{{0.5}}$

To apply the aforementioned formula for binomial series, we may replace " $\displaystyle{x}$ " with " $\displaystyle{{x}}^{{2}}$ " and " $\displaystyle{k}$ " with " $\displaystyle{0.5}$ ". We let:

$\displaystyle{{\left({1}+{{x}}^{{2}}\right)}}^{{0.5}}={\sum_{{{n}={0}}}^{\infty}}\frac{{{0.5}{\left({0.5}-{1}\right)}{\left({0.5}-{2}\right)}\ldots{\left({0.5}-{n}+{1}\right)}}}{{{n}!}}{{\left({{x}}^{{2}}\right)}}^{{n}}$

$\displaystyle={\sum_{{{n}={0}}}^{\infty}}\frac{{{0.5}{\left({0.5}-{1}\right)}{\left({0.5}-{2}\right)}\ldots{\left({0.5}-{n}+{1}\right)}}}{{{n}!}}{{x}}^{{{2}{n}}}$

$\displaystyle={1}+{0.5}{{x}}^{{{2}\cdot{1}}}+\frac{{{0.5}{\left({0.5}-{1}\right)}}}{{{2}!}}{{x}}^{{{2}\cdot{2}}}+\frac{{{0.5}{\left({0.5}-{1}\right)}{\left({0.5}-{2}\right)}}}{{{3}!}}{{x}}^{{{2}\cdot{3}}}+\frac{{{0.5}{\left({0.5}-{1}\right)}{\left({0.5}-{2}\right)}{\left({0.5}-{3}\right)}}}{{{4}!}}{{x}}^{{{2}\cdot{4}}}+\ldots$

$\displaystyle={1}+{0.5}{{x}}^{{2}}-\frac{{0.25}}{{{1}\cdot{2}}}{{x}}^{{4}}+\frac{{0.375}}{{{1}\cdot{2}\cdot{3}}}{{x}}^{{6}}-\frac{{0.9375}}{{{1}\cdot{2}\cdot{3}\cdot{4}}}{{x}}^{{8}}+\ldots$

$\displaystyle={1}+{0.5}{{x}}^{{2}}-\frac{{0.25}}{{2}}{{x}}^{{4}}+\frac{{0.375}}{{6}}{{x}}^{{6}}-\frac{{0.9375}}{{24}}{{x}}^{{8}}+\ldots$

$\displaystyle={1}+\frac{{{x}}^{{2}}}{{2}}-\frac{{{x}}^{{4}}}{{8}}+\frac{{{x}}^{{6}}}{{16}}-\frac{{{5}{{x}}^{{8}}}}{{128}}+\ldots$

Thus, the Maclaurin series for the $\displaystyle{f{{\left({x}\right)}}}=\sqrt{{{1}+{{x}}^{{2}}}}$ can be expressed as:

$\displaystyle\sqrt{{{1}+{{x}}^{{2}}}}={1}+\frac{{{x}}^{{2}}}{{2}}-\frac{{{x}}^{{4}}}{{8}}+\frac{{{x}}^{{6}}}{{16}}-\frac{{{5}{{x}}^{{8}}}}{{128}}+\ldots$

New Notes Blog

Sunday, 11 August 2013

$\displaystyle{f{{\left({x}\right)}}}=\sqrt{{{1}+{{x}}^{{2}}}}$ Use the binomial series to find the Maclaurin series for the function.

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In "By the Waters of Babylon," under the leadership of John, what do you think the Hill People will do with their society?

Sunday, 11 August 2013

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

No comments:

Post a Comment

In &quot;By the Waters of Babylon,&quot; under the leadership of John, what do you think the Hill People will do with their society?

$\displaystyle{f{{\left({x}\right)}}}=\sqrt{{{1}+{{x}}^{{2}}}}$ Use the binomial series to find the Maclaurin series for the function.

In "By the Waters of Babylon," under the leadership of John, what do you think the Hill People will do with their society?