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2^n/n!

Sum of series 2^n/n!



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The solution

You have entered [src]
  oo    
____    
\   `   
 \     n
  \   2 
  /   --
 /    n!
/___,   
n = 1   
$$\sum_{n=1}^{\infty} \frac{2^{n}}{n!}$$
Sum(2^n/factorial(n), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{2^{n}}{n!}$$
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
$$a_{n} = \frac{1}{n!}$$
and
$$x_{0} = -2$$
,
$$d = 1$$
,
$$c = 0$$
then
$$R = \tilde{\infty} \left(-2 + \lim_{n \to \infty} \left|{\frac{\left(n + 1\right)!}{n!}}\right|\right)$$
Let's take the limit
we find
$$R = \infty$$
The rate of convergence of the power series
The answer [src]
      2
-1 + e 
$$-1 + e^{2}$$
-1 + exp(2)
Numerical answer [src]
6.38905609893065022723042746058
6.38905609893065022723042746058
The graph
Sum of series 2^n/n!

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