Chapter 9 - Section 9.10 - The Binomial Series and Applications of Taylor Series - Exercises - Page 550: 50

$$ x^2 e^{-2x}$$

The Taylor series for $ e^x $ can be defined as: $ e^x=1+x+\dfrac{x^2}{2!}+.....\dfrac{x^n}{n!}...; |x| \lt \infty $ Consider the given series: $\\=x^2-2x^3+\dfrac{2^2 x^4}{2!}-\dfrac{x^2}{2!}-\dfrac{2^3 x^5}{3!}...\\=x^2(1-2x+\dfrac{(2x)^2}{2!}-\dfrac{(2x)^3}{3!}+....)\\=x^2 e^{-2x}$