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(3*n+1)/factorial(n)
Sum of series/ (3*n+1)/factorial(n)

Sum of series (3*n+1)/factorial(n)


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The solution

You have entered [src] 
  oo         
 ___         
 \  `        
  \   3*n + 1
   )  -------
  /      n!  
 /__,        
n = 1
$$\sum_{n=1}^{\infty} \frac{3 n + 1}{n!}$$ 
Sum((3*n + 1)/factorial(n), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{3 n + 1}{n!}$$
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
$$a_{n} = \frac{3 n + 1}{n!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(3 n + 1\right) \left|{\frac{\left(n + 1\right)!}{n!}}\right|}{3 n + 4}\right)$$
Let's take the limit
we find
False

False
The rate of convergence of the power series
The answer [src] 
-1 + 4*E
$$-1 + 4 e$$ 
-1 + 4*E
Numerical answer [src] 
9.87312731383618094144114988541
9.87312731383618094144114988541
The graph
Sum of series (3*n+1)/factorial(n)

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