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.

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

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

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.

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

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

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