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(3^n-1)/factorial(n+1)
Sum of series/ (3^n-1)/factorial(n+1)

Sum of series (3^n-1)/factorial(n+1)


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The solution

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  oo          
____          
\   `         
 \      n     
  \    3  - 1 
  /   --------
 /    (n + 1)!
/___,         
n = 1
∑n=1∞3n−1(n+1)!\sum_{n=1}^{\infty} \frac{3^{n} - 1}{\left(n + 1\right)!} 
Sum((3^n - 1)/factorial(n + 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
3n−1(n+1)!\frac{3^{n} - 1}{\left(n + 1\right)!}
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−1(n+1)!a_{n} = \frac{3^{n} - 1}{\left(n + 1\right)!}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=lim⁡n→∞∣(3n−1)(n+2)!(3n+1−1)(n+1)!∣1 = \lim_{n \to \infty} \left|{\frac{\left(3^{n} - 1\right) \left(n + 2\right)!}{\left(3^{n + 1} - 1\right) \left(n + 1\right)!}}\right|
Let's take the limit
we find
False

False
The rate of convergence of the power series
1.07.01.52.02.53.03.54.04.55.05.56.06.505
The answer [src] 
         3
2       e 
- - E + --
3       3
−e+23+e33- e + \frac{2}{3} + \frac{e^{3}}{3} 
2/3 - E + exp(3)/3
Numerical answer [src] 
4.64356381260351067828255574684
4.64356381260351067828255574684
The graph
Sum of series (3^n-1)/factorial(n+1)

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