Question

The coefficient of the third term in the Taylor series of $(x-1)e^x$ is?

• A. $\frac{1}{3}$
• B. $3$
• C. $2$
• D. $\frac{1}{2}$

Answer 1

The correct option is A $\frac{1}{3}$

We know that taylor series of $e^x$ is given by

$e^x=1+x^1+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}...$

$(x-1)e^x=(x-1)(1+x^1+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}...)$

$=-1+\frac{1}{2}{x}^2+\frac{1}{3}{x}^3+\frac{1}{8}{x}^4+\frac{1}{30}{x}^5+....$

Hence, the cofficient of the third term in a taylor series of $(x-1)e^x$ is $\frac{1}{3}$