Maclaurin series expension.
[katex]\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+……}[/katex]
The expension of [katex]f(x)[/katex] is called the Maclaurin series expension.
The above expansion is called Maclaurin Theorem.
Example
[katex]f(x)=a^x[/katex]
Apply Maclaurin series expension.
[katex]f'(x)=a^x\ln x[/katex]
Add title
[katex]\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+……}[/katex]
The expension of [katex]f(x)[/katex] is called the Maclaurin series expension.
The above expansion is called Maclaurin Theorem.
Example
[katex]f(x)=a^x[/katex]
Apply Maclaurin series expension.
[katex]f'(x)=a^x(\ln x)[/katex]
[katex]f”(x)=a^x(\ln x)^2[/katex]
[katex]f”'(x)=a^x(\ln x)^3[/katex]
[katex]f^{iv}(x)=a^x(\ln x)^4[/katex]
Put [katex]x=0[/katex] above equations
[katex]f(0)=a^0[/katex]
[katex]\boxed{f(0)=1}[/katex]
[katex]f'(0)=a^0(\ln x)[/katex]
[katex]\boxed{f'(0)=(\ln x)}[/katex]
[katex]f”(0)=a^0(\ln x)^2[/katex]
[katex]\boxed{f”(0)=(\ln x)^2}[/katex]
[katex]f”'(0)=a^0(\ln x)^3[/katex]
[katex]\boxed{f”'(0)=(\ln x)^3}[/katex]
[katex]f^{iv}(0)=a^0(\ln x)^4[/katex]
[katex]\boxed{f^{iv}(0)=(\ln x)^4}[/katex]
Substituting these values in the formula.
[katex]\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+……}[/katex]
[katex]\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}[/katex]
Example
[katex]y=x^n[/katex] [katex]n\neq0[/katex]
Apply Maclaurin series expension.
[katex]y’=nx^{n-1}[/katex]
[katex]y”=n(n-1)x^{n-2}[/katex]
[katex]y”’=n(n-1)(n-2)x^{n-3}[/katex]
[katex]y^{iv}=n(n-1)(n-2)(n-3)x^{n-4}[/katex]
Put [katex]x=0[/katex] above equations
[katex]y(0)=0^n[/katex]
[katex]\boxed{y(0)=1}[/katex]
[katex]y'(0)=n0^{n-1}[/katex]
[katex]\boxed{y'(0)=n}[/katex]
[katex]y”(0)=n(n-1)0^{n-2}[/katex]
[katex]\boxed{y”(0)=n(n-1)}[/katex]
[katex]y”'(0)=n(n-1)(n-2)0^{n-3}[/katex]
[katex]\boxed{y”'(0)=n(n-1)(n-2)}[/katex]
[katex]y^{iv}(0)=n(n-1)(n-2)(n-3)0^{n-4}[/katex]
[katex]\boxed{y^{iv}(0)=n(n-1)(n-2)(n-3)}[/katex]
Substituting these values in the formula.
[katex]\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+……}[/katex]
[katex]\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!}…}[/katex]