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.
Examplef(x)=a^xApply Maclaurin series expension.
f'(x)=a^x\ln xAdd title
\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)^2f'''(x)=a^x(\ln x)^3f^{iv}(x)=a^x(\ln x)^4Put 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}
Exampley=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!}…}
