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

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

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  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)

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