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e^(-n)

Sum of series e^(-n)



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The solution

You have entered [src]
  oo     
 ___     
 \  `    
  \    -n
  /   E  
 /__,    
n = 1    
n=1en\sum_{n=1}^{\infty} e^{- n}
Sum(E^(-n), (n, 1, oo))
The radius of convergence of the power series
Given number:
ene^{- n}
It is a series of species
an(cxx0)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+limnanan+1cR^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}
In this case
an=1a_{n} = 1
and
x0=ex_{0} = - e
,
d=1d = -1
,
c=0c = 0
then
1R=~(e+limn1)\frac{1}{R} = \tilde{\infty} \left(- e + \lim_{n \to \infty} 1\right)
Let's take the limit
we find
False

R=0R = 0
The rate of convergence of the power series
1.07.01.52.02.53.03.54.04.55.05.56.06.50.250.75
The answer [src]
   -1  
  e    
-------
     -1
1 - e  
1e(1e1)\frac{1}{e \left(1 - e^{-1}\right)}
exp(-1)/(1 - exp(-1))
Numerical answer [src]
0.581976706869326424385002005109
0.581976706869326424385002005109
The graph
Sum of series e^(-n)

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