Mister Exam

Sum of series exp(-n)



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The solution

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

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

    Examples of finding the sum of a series