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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

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  oo         
 ___         
 \  `        
  \   3*n + 1
   )  -------
  /      n!  
 /__,        
n = 1
∑n=1∞3n+1n!\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:
3n+1n!\frac{3 n + 1}{n!}
It is a series of species
an(cx−x0)dna_{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:
Rd=x0+lim⁡n→∞∣anan+1∣cR^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}
In this case
an=3n+1n!a_{n} = \frac{3 n + 1}{n!}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=lim⁡n→∞((3n+1)∣(n+1)!n!∣3n+4)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
1.07.01.52.02.53.03.54.04.55.05.56.06.5010
The answer [src] 
-1 + 4*E
−1+4e-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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