Double Math

# Maclaurin series expension with examples

Maclaurin series expension.

\boxed{f\left(x\right)=f\left(0\right)+f'\left(0\right)x+\frac{f''\left(0\right)}{2!}x^2+\frac{f'''\left(0\right)}{3!}x^3+\frac{f^{iv}\left(0\right)}{4!}x^4+……}

The expension of f(x) is called the Maclaurin series expension.

The above expansion is called Maclaurin Theorem.

Example
f(x)=a^x

Apply Maclaurin series expension.

f'(x)=a^x\ln x

\boxed{f\left(x\right)=f\left(0\right)+f'\left(0\right)x+\frac{f''\left(0\right)}{2!}x^2+\frac{f'''\left(0\right)}{3!}x^3+\frac{f^{iv}\left(0\right)}{4!}x^4+……}

The expension of f(x) is called the Maclaurin series expension.

The above expansion is called Maclaurin Theorem.

Example

f(x)=a^x

Apply Maclaurin series expension.

f'(x)=a^x(\ln x)

f''(x)=a^x(\ln x)^2

f'''(x)=a^x(\ln x)^3

f^{iv}(x)=a^x(\ln x)^4

Put x=0 above equations

f(0)=a^0

\boxed{f(0)=1}

f'(0)=a^0(\ln x)

\boxed{f'(0)=(\ln x)}

f''(0)=a^0(\ln x)^2

\boxed{f''(0)=(\ln x)^2}

f'''(0)=a^0(\ln x)^3

\boxed{f'''(0)=(\ln x)^3}

f^{iv}(0)=a^0(\ln x)^4

\boxed{f^{iv}(0)=(\ln x)^4}

Substituting these values in the formula.

\boxed{f\left(x\right)=f\left(0\right)+f'\left(0\right)x+\frac{f''\left(0\right)}{2!}x^2+\frac{f'''\left(0\right)}{3!}x^3+\frac{f^{iv}\left(0\right)}{4!}x^4+……}

\boxed{a^x=1+\ln a+\frac{\left(\ln\;a\right)^2}{2!}x^2+\frac{\left(\ln\;a\right)^3}{3!}x^3+\frac{\left(\ln\;a\right)^4}{4!}x^4+\dots\dots}

Example

y=x^n n\neq0

Apply Maclaurin series expension.

y'=nx^{n-1}

y''=n(n-1)x^{n-2}

y'''=n(n-1)(n-2)x^{n-3}

y^{iv}=n(n-1)(n-2)(n-3)x^{n-4}

Put x=0 above equations

y(0)=0^n

\boxed{y(0)=1}

y'(0)=n0^{n-1}

\boxed{y'(0)=n}

y''(0)=n(n-1)0^{n-2}

\boxed{y''(0)=n(n-1)}

y'''(0)=n(n-1)(n-2)0^{n-3}

\boxed{y'''(0)=n(n-1)(n-2)}

y^{iv}(0)=n(n-1)(n-2)(n-3)0^{n-4}

\boxed{y^{iv}(0)=n(n-1)(n-2)(n-3)}

Substituting these values in the formula.

\boxed{f\left(x\right)=f\left(0\right)+f'\left(0\right)x+\frac{f''\left(0\right)}{2!}x^2+\frac{f'''\left(0\right)}{3!}x^3+\frac{f^{iv}\left(0\right)}{4!}x^4+……}

\boxed{x^n=1+nx+\frac{n(n-1)x^2}{2!}+\frac{n(n-1)(n-2)x^3}{3!}+\frac{n(n-1)(n-2)(n-4)x^4}{4!}…}