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n*exp(-n^2)
Sum of series/ n*exp(-n^2)

Sum of series n*exp(-n^2)


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The solution

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  oo        
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 \  `       
  \        2
   )     -n 
  /   n*e   
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n = 1
$$\sum_{n=1}^{\infty} n e^{- n^{2}}$$ 
Sum(n*exp(-n^2), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$n e^{- n^{2}}$$
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
$$a_{n} = n e^{- n^{2}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{n e^{- n^{2}} e^{\left(n + 1\right)^{2}}}{n + 1}\right)$$
Let's take the limit
we find
False

False
The rate of convergence of the power series
Numerical answer [src] 
0.404881398571310708910099048243
0.404881398571310708910099048243
The graph
Sum of series n*exp(-n^2)

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