Use a known Taylor series to find the Taylor series about c = 0 for , and find its radius of convergence.

Question

Use a known Taylor series to find the Taylor series about c = 0 for  , and find its radius of convergence.

Quizplus · Accepted Answer

We know that the Taylor series for $e^x$ about $c=0$ is:
$$e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}$$
If we substitute $x=11eaa88b_{90fe_{32b3}a696_{31e914076f84}}^{TB5869_{11}}$, we get:
$$11eaa88b_{90fe_{32b3}a696_{31e914076f84}}^{TB5869_{11}}=\sum_{n=0}^{\infty}\frac{(11eaa88b_{90fe_{32b3}a696_{31e914076f84}}^{TB5869_{11}})^n}{n!}$$
Therefore, the Taylor series for $11eaa88b_{90fe_{32b3}a696_{31e914076f84}}^{TB5869_{11}}$ about $c=0$ has the same coefficients as the Taylor series for $e^x$ about $c=0$, except with $x$ replaced by $11eaa88b_{90fe_{32b3}a696_{31e914076f84}}^{TB5869_{11}}$.

The radius of convergence of the Taylor series for $e^x$ is infinite, so the radius of convergence of the Taylor series for $11eaa88b_{90fe_{32b3}a696_{31e914076f84}}^{TB5869_{11}}$ is also infinite.

Therefore, the best choice is B.

Use a Known Taylor Series to Find the Taylor Series