Question

The third term in the Taylor series expansion of $e^x$ about x=1 would be

• A. $e^a(x-a)$
• B. $\frac{e^a}{2}(x-a{)}^2$
• C. $\frac{x^2}{2}$
  
• D. $\frac{e^a}{6}(x-a{)}^3$

Answer 1

The correct option is B $\frac{e^a}{2}(x-a{)}^2$

$f\left(x\right)=e^x=e^{x-a+a}=e^a{e}^{x-a}$
$=e^a[1+(x-a)+\frac{(x-a{)}^2}{2!}+...]$
Third term=$\frac{e^a(x-a{)}^2}{2}$