Find a power series for the function centered at 1.

Question

Find a power series for the function  centered at 1. ​

Quizplus · Accepted Answer

The power series centered at 1 is of the form $\sum_{n=0}^{\infty}c_n(x-1)^n$, where $c_n=\frac{f^{(n)}(1)}{n!}$ and $f(x)=11e^x$. Taking derivatives of $f(x)$, we get $f^{(n)}(x)=11e^x$, so $c_n=\frac{11e}{n!}$ and the power series is $\sum_{n=0}^{\infty}\frac{11e}{n!}(x-1)^n = 11e\sum_{n=0}^{\infty}\frac{(x-1)^n}{n!}$. This is just the power series expansion of $11e^{x-1}$, so the best choice is C.

Find a Power Series for the Function Centered at 1