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(3^n-1)/n!

Sum of series (3^n-1)/n!



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The solution

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  oo        
____        
\   `       
 \     n    
  \   3  - 1
  /   ------
 /      n!  
/___,       
n = 0       
$$\sum_{n=0}^{\infty} \frac{3^{n} - 1}{n!}$$
Sum((3^n - 1)/factorial(n), (n, 0, 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
$$1 = \lim_{n \to \infty} \left|{\frac{\left(3^{n} - 1\right) \left(n + 1\right)!}{\left(3^{n + 1} - 1\right) n!}}\right|$$
Let's take the limit
we find
False

False
The rate of convergence of the power series
The answer [src]
      3
-E + e 
$$- e + e^{3}$$
-E + exp(3)
Numerical answer [src]
17.3672550947286225055682421832
17.3672550947286225055682421832
The graph
Sum of series (3^n-1)/n!

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