Approximating powers Compute the coefficients for the Taylor series for the following functions about the given point a, and then use the first four terms of the series to approximate the given number. f(x)=frac{1}{sqrt{x}} with a=4, approximate frac{1}{sqrt{3}}

Question
Series
asked 2021-02-05
Approximating powers Compute the coefficients for the Taylor series for the following functions about the given point a, and then use the first four terms of the series to approximate the given number.
\(\displaystyle{f{{\left({x}\right)}}}={\frac{{{1}}}{{\sqrt{{{x}}}}}}\) with \(\displaystyle{a}={4}\), approximate \(\displaystyle{\frac{{{1}}}{{\sqrt{{{3}}}}}}\)

Answers (1)

2021-02-06
Consider the provided function,
\(\displaystyle{f{{\left({x}\right)}}}={\frac{{{1}}}{{\sqrt{{{x}}}}}}\) with \(\displaystyle{a}={4}\), approximate \(\displaystyle{\frac{{{1}}}{{\sqrt{{{3}}}}}}\)
Compute the coefficients for the Taylor series for the following functions about the given point a, and then use the first four terms of the series to approximate the given number.
The Taylor series as a form of \(\displaystyle{\sum_{{{k}={0}}}^{\infty}}{c}_{{k}}{\left({x}-{a}\right)}^{{k}}\), where \(\displaystyle{c}_{{k}}={\frac{{{{f}^{{k}}{\left({a}\right)}}}}{{{k}!}}},{k}={0},{1},{2},\ldots\)
We evaluate the derivative of the function at point \(\displaystyle{a}={4}\)
\(\displaystyle{f{{\left({x}\right)}}}={\frac{{{1}}}{{\sqrt{{{x}}}}}}\Rightarrow{f{{\left({4}\right)}}}={\frac{{{1}}}{{{2}}}}\)
\(\displaystyle{f}'{\left({x}\right)}=-{\frac{{{1}}}{{{2}{x}^{{{\frac{{{3}}}{{{2}}}}}}}}}\Rightarrow{f}'{\left({4}\right)}=-{\frac{{{1}}}{{{16}}}}\)
\(\displaystyle{f}{''}{\left({x}\right)}={\frac{{{3}}}{{{4}{x}^{{{\frac{{{5}}}{{{2}}}}}}}}}\Rightarrow{f}{''}{\left({4}\right)}={\frac{{{3}}}{{{128}}}}\)
\(\displaystyle{f}{'''}{\left({x}\right)}=-{\frac{{{15}}}{{{8}{x}^{{{\frac{{{7}}}{{{2}}}}}}}}}\Rightarrow{f}{'''}{\left({4}\right)}=-{\frac{{{15}}}{{{1024}}}}\)
Hence, the first four term of the series is shown below.
\(\displaystyle{\frac{{{1}}}{{{2}}}}-{\frac{{{1}}}{{{16}}}}{\left({x}-{4}\right)}+{\frac{{{3}}}{{{128}}}}{\left({x}-{4}\right)}^{{2}}-{\frac{{{15}}}{{{1024}}}}{\left({x}-{4}\right)}^{{3}}\)
So, \(\displaystyle{\frac{{{1}}}{{\sqrt{{{3}}}}}}={\frac{{{1}}}{{{2}}}}-{\frac{{{1}}}{{{16}}}}{\left({\frac{{{1}}}{{\sqrt{{{3}}}}}}-{4}\right)}+{\frac{{{3}}}{{{128}}}}{\left({\frac{{{1}}}{{\sqrt{{{3}}}}}}-{4}\right)}^{{2}}-{\frac{{{15}}}{{{1024}}}}{\left({\frac{{{1}}}{{\sqrt{{{3}}}}}}-{4}\right)}^{{3}}\)
\(\displaystyle={\frac{{{1}}}{{{2}}}}-{\frac{{{1}-{4}\sqrt{{{3}}}}}{{{16}\sqrt{{{3}}}}}}+{\frac{{{49}\sqrt{{{3}}}-{24}}}{{{128}\sqrt{{{3}}}}}}-{\frac{{{5}{\left({145}-{204}\sqrt{{{3}}}\right)}}}{{{1024}\sqrt{{{3}}}}}}\)
\(\displaystyle={\frac{{{1}}}{{{2}}}}+{\frac{{{1668}-{327}\sqrt{{{3}}}}}{{{1024}}}}\)
Hence.
0

Relevant Questions

asked 2020-12-30
Approximating powers Compute the coefficients for the Taylor series for the following functions about the given point a, and then use the first four terms of the series to approximate the given number.
\(f(x)=\frac{1}{\sqrt{x}}\) with \(a=4\), approximate \(\frac{1}{\sqrt{3}}\)
asked 2021-02-05
Any method
a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients.
b. Determine the radius of convergence of the series.
\(f(x)=\cos2x+2\sin x\)
asked 2020-11-22
Any method
a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients.
\(f(x)=x^2\cos x^2\)
asked 2020-12-30
Taylor series Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.
\(f(x)=\cosh(2x-2),a=1\)
asked 2021-01-02
Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.
\(f(x)=\tan^{-1}4x,a=0\)
asked 2020-12-29
Taylor series Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.
\(\displaystyle{f{{\left({x}\right)}}}={\text{cosh}{{\left({2}{x}-{2}\right)}}},{a}={1}\)
asked 2020-12-01
Binomial series
a. Find the first four nonzero terms of the binomial series centered at 0 for the given function.
b. Use the first four terms of the series to approximate the given quantity.
\(f(x)=(1+x)^{\frac{2}{3}}\), approximate \((1.02)^{\frac{2}{3}}\)
asked 2021-03-11
Taylor series and interval of convergence
a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.
b. Write the power series using summation notation.
c. Determine the interval of convergence of the series.
\(f(x)=\log_3(x+1),a=0\)
asked 2021-02-25
Taylor series
a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a.
b. Write the power series using summation notation.
\(f(x)=x\ln x-x+1,a=1\)
asked 2021-01-19
Taylor series
a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a.
b. Write the power series using summation notation.
\(f(x)=2^x,a=1\)
...