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e^n/n^10

Sum of series e^n/n^10



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The solution

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  oo     
____     
\   `    
 \      n
  \    E 
   )  ---
  /    10
 /    n  
/___,    
n = 1    
n=1enn10\sum_{n=1}^{\infty} \frac{e^{n}}{n^{10}}
Sum(E^n/n^10, (n, 1, oo))
The radius of convergence of the power series
Given number:
enn10\frac{e^{n}}{n^{10}}
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=1n10a_{n} = \frac{1}{n^{10}}
and
x0=ex_{0} = - e
,
d=1d = 1
,
c=0c = 0
then
R=~(e+limn((n+1)10n10))R = \tilde{\infty} \left(- e + \lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{10}}{n^{10}}\right)\right)
Let's take the limit
we find
False
The rate of convergence of the power series
1.07.01.52.02.53.03.54.04.55.05.56.06.52.7102.730
The answer [src]
oo
\infty
oo
The graph
Sum of series e^n/n^10

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