Q 29 - Calculus

Question

Use the Maclaurin series for $e^x$to find power series representations

for $g\left(x\right)=x^2{e}^x,g^{\text{'}}\left(x\right)$ and $\int g\left(x\right)dx$

Step-by-Step Solution

Answer

The power series representation for $g\left(x\right)=x^2{e}^x$is $g\left(x\right)=\sum _{k=0}^{\infty }\frac{1}{k!}{x}^{k+2}$

Step 1: Given information

The function is $f\left(x\right)=e^x$

Step 2: Find the Maclaurin series for the function

The Maclaurin series for the function $f\left(x\right)=e^x$is

$1+x+\frac{1}{2!}{x}^2+\frac{1}{3!}{x}^3+\frac{1}{4!}{x}^4+\dots$

Or $f\left(x\right)=\sum _{k=0}^{\infty }\frac{1}{k!}{x}^k$

Hence, $g\left(x\right)=x^2{e}^x$

$$\begin{gathered} g\left(x\right)=x^2\sum _{k=0}^8\frac{1}{k!}{x}^k \\ =\sum _{k=0}^8\frac{1}{k!}{x}^{k+2} \end{gathered}$$

Step 3: Find the power series representation for g ( x ) = x 2 e x  

The power series representation for $g\left(x\right)=x^2{e}^x$is

$g\left(x\right)=x^2\sum _{k=0}^{\infty }\frac{1}{k!}{x}^k$

$g\left(x\right)=\sum _{k=0}^{\infty }\frac{1}{k!}{x}^{k+2}$