Question

8) Write the first four terms of a Maclaurin series for 𝑓(𝑥)=𝑒𝑥?

Answer 1

The Taylor series of a function f (x) that is infinitely differentiable at a number $a$  is the power series

$f(x)=f(a)+\frac{f'(a)}{1!}(x-a)+\frac{f''(a)}{2!}(x-a)^2+$$\frac{f'''(a)}{3!}(x-a)^2+...$ .

When $a=0$, the series is also called a Maclaurin series.

$f(x)=e^x$ , $f(0)=1$ ;

$f'(x)=e^x$ , $f'(0)=1$ ;

$f''(x)=e^x$ , $f''(0)=1$ ;

$f'''(x)=e^x$ , $f'''(0)=1$ .

$e^x=1+\frac{1}{1!}(x-0)+\frac{1}{2!}(x-0)^2+\frac{1}{3!}(x-0)^3+...=$$1+x+\frac12x^2+\frac16x^3+...$

Answer: $e^x=1+x+\frac12x^2+\frac16x^3+...$ .