Sum of series exp(nx)

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  oo      
 ___      
 \  `     
  \    n*x
  /   e   
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n = 0

\sum_{n=0}^{\infty} e^{n x}

Sum(exp(n*x), (n, 0, oo))

The radius of convergence of the power series

Given number:

e^{n x}

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} = e^{n x}

and

x_{0} = 0

d = 0

c = 1

then

1 = \lim_{n \to \infty}\left(e^{n \operatorname{re}{\left(x\right)}} e^{- \left(n + 1\right) \operatorname{re}{\left(x\right)}}\right)

1 = e^{- \operatorname{re}{\left(x\right)}}

False