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e^-sqrt(n)/sqrt(n)
Sum of series/ e^-sqrt(n)/sqrt(n)

Sum of series e^-sqrt(n)/sqrt(n)


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The solution

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

False
The rate of convergence of the power series
The answer [src] 
  oo          
_____         
\    `        
 \         ___
  \     -\/ n 
   \   e      
   /   -------
  /       ___ 
 /      \/ n  
/____,        
n = 1
$$\sum_{n=1}^{\infty} \frac{e^{- \sqrt{n}}}{\sqrt{n}}$$ 
Sum(exp(-sqrt(n))/sqrt(n), (n, 1, oo))
Numerical answer [src] 
0.948539669193155639410263611681
0.948539669193155639410263611681
The graph
Sum of series e^-sqrt(n)/sqrt(n)

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