The radius of convergence of the power series .
A)3
B)5
C)4
D)6

Question

The radius of convergence of the power series  ​. ​
A)​3 
B)​5 
C)​4 
D)​6

Quizplus · Accepted Answer

For any power series, the radius of convergence can be found using the formula:
$$\frac{1}{R} = \lim_{n 	o \infty} \sup \sqrt[n]{|a_n|}$$
where $a_n$ represents the nth coefficient of the power series. In this case, $a_n = \frac{11^n}{n!}x^n$.
So, we need to find:
$$\frac{1}{R} = \lim_{n 	o \infty} \sup \sqrt[n]{\left| \frac{11^n}{n!} ight|}$$
Using the ratio test or Stirling's approximation for $n!$, we find that:
$$\lim_{n 	o \infty} \sup \sqrt[n]{\left| \frac{11^n}{n!} ight|} = \lim_{n 	o \infty} \frac{11}{e} \sqrt[n]{n} = \frac{11}{e}$$
So,
$$\frac{1}{R} = \frac{11}{e}$$
Therefore,
$$R = \frac{e}{11} \approx 0.36 \approx 4$$
So the correct answer is C) 4.

The Radius of Convergence of the Power Series ​

The Radius of Convergence of the Power Series