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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

You have entered [src] 
  oo          
____          
\   `         
 \      n + 1 
  \    2      
  /   --------
 /    (n + 1)!
/___,         
n = 1
$$\sum_{n=1}^{\infty} \frac{2^{n + 1}}{\left(n + 1\right)!}$$ 
Sum(2^(n + 1)/factorial(n + 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{2^{n + 1}}{\left(n + 1\right)!}$$
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
$$a_{n} = \frac{2^{n + 1}}{\left(n + 1\right)!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(2^{- n - 2} \cdot 2^{n + 1} \left|{\frac{\left(n + 2\right)!}{\left(n + 1\right)!}}\right|\right)$$
Let's take the limit
we find
False

False
The rate of convergence of the power series
The answer [src] 
      2
-3 + e
$$-3 + e^{2}$$ 
-3 + exp(2)
Numerical answer [src] 
4.38905609893065022723042746058
4.38905609893065022723042746058
The graph
Sum of series 2^(n+1)/factorial(n+1)

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