Mister Exam

Lang:

Sum of series/ 3^n/n!

Sum of series 3^n/n!

You have entered [src]

  oo    
____    
\   `   
 \     n
  \   3 
  /   --
 /    n!
/___,   
n = 1

\sum_{n=1}^{\infty} \frac{3^{n}}{n!}

Sum(3^n/factorial(n), (n, 1, oo))

The radius of convergence of the power series

Given number:

\frac{3^{n}}{n!}

It is a series of species

a_{n} \left(c x - x_{0}\right)^{d n}

- power series.
The radius of convergence of a power series can be calculated by the formula:

R^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}

In this case

a_{n} = \frac{1}{n!}

and

x_{0} = -3

d = 1

c = 0

then

R = \tilde{\infty} \left(-3 + \lim_{n \to \infty} \left|{\frac{\left(n + 1\right)!}{n!}}\right|\right)

R = \infty

The rate of convergence of the power series

The answer [src]

      3
-1 + e

-1 + e^{3}

-1 + exp(3)

Numerical answer [src]

19.0855369231876677409285296546

19.0855369231876677409285296546

The graph