Find a power series for the function centered at 0.

Question

Find a power series for the function  centered at 0. ​

Quizplus · Accepted Answer

The power series for the function centered at 0 is given by $\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n$. Since the function is $e^{x^2}$, we can use the Maclaurin series for $e^x$ and replace $x$ with $x^2$: $e^{x^2}=\sum_{n=0}^{\infty} \frac{(x^2)^n}{n!}=\sum_{n=0}^{\infty} \frac{x^{2n}}{n!}$. Therefore, we have $f^{(n)}(x) = \frac{d^n}{dx^n}(e^{x^2}) = 2^ne^{x^2}x^{2n-1}$ and $f^{(n)}(0) = 0$ for odd $n$ and $2^n$ for even $n$. So the power series for the function centered at 0 is $\sum_{n=0}^{\infty} \frac{2^nx^{2n}}{(2n)!}$. Plugging in $x=1$ gives us $\sum_{n=0}^{\infty} \frac{2^n}{(2n)!} = 11.896$. The only choice close to this value is C, which is 11eac44f_ec21_cd53_bedf_0bc786a70be5_TB7497_11.

Find a Power Series for the Function Centered at 0