Mister Exam

Sum of series exp(nx)



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The solution

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  oo      
 ___      
 \  `     
  \    n*x
  /   e   
 /__,     
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
$$1 = \lim_{n \to \infty}\left(e^{n \operatorname{re}{\left(x\right)}} e^{- \left(n + 1\right) \operatorname{re}{\left(x\right)}}\right)$$
Let's take the limit
we find
$$1 = e^{- \operatorname{re}{\left(x\right)}}$$
False

    Examples of finding the sum of a series