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exp(n*(-3)/2)
Sum of series/ exp(n*(-3)/2)

Sum of series exp(n*(-3)/2)


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The solution

You have entered [src] 
  oo         
____         
\   `        
 \     n*(-3)
  \    ------
  /      2   
 /    e      
/___,        
n = 1
$$\sum_{n=1}^{\infty} e^{\frac{\left(-3\right) n}{2}}$$ 
Sum(exp((n*(-3))/2), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$e^{\frac{\left(-3\right) n}{2}}$$
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
$$a_{n} = 1$$
and
$$x_{0} = - e$$
,
$$d = - \frac{3}{2}$$
,
$$c = 0$$
then
$$\frac{1}{R^{\frac{3}{2}}} = \tilde{\infty} \left(- e + \lim_{n \to \infty} 1\right)$$
Let's take the limit
we find
False

$$R = 0$$
The rate of convergence of the power series
The answer [src] 
   -3/2  
  e      
---------
     -3/2
1 - e
$$\frac{1}{\left(1 - e^{- \frac{3}{2}}\right) e^{\frac{3}{2}}}$$ 
exp(-3/2)/(1 - exp(-3/2))
Numerical answer [src] 
0.287216916788868244336781610543
0.287216916788868244336781610543
The graph
Sum of series exp(n*(-3)/2)

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