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2^(n+1)/factorial(n-1)
Sum of series/ 2^(n+1)/factorial(n-1)

Sum of series 2^(n+1)/factorial(n-1)


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The solution

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

False
The rate of convergence of the power series
3.09.03.54.04.55.05.56.06.57.07.58.08.5020
The answer [src] 
         2
-12 + 4*e
−12+4e2-12 + 4 e^{2} 
-12 + 4*exp(2)
Numerical answer [src] 
17.5562243957226009089217098423
17.5562243957226009089217098423
The graph
Sum of series 2^(n+1)/factorial(n-1)

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