Mister Exam

Sum of series exp(nx)



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The solution

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  oo      
 ___      
 \  `     
  \    n*x
  /   e   
 /__,     
n = 0     
n=0enx\sum_{n=0}^{\infty} e^{n x}
Sum(exp(n*x), (n, 0, oo))
The radius of convergence of the power series
Given number:
enxe^{n x}
It is a series of species
an(cxx0)dna_{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:
Rd=x0+limnanan+1cR^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}
In this case
an=enxa_{n} = e^{n x}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=limn(enre(x)e(n+1)re(x))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=ere(x)1 = e^{- \operatorname{re}{\left(x\right)}}
False

    Examples of finding the sum of a series