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Sum of series/ e^n/n^10

Sum of series e^n/n^10

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  oo     
____     
\   `    
 \      n
  \    E 
   )  ---
  /    10
 /    n  
/___,    
n = 1

\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:

\frac{e^{n}}{n^{10}}

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} = \frac{1}{n^{10}}

and

x_{0} = - e

d = 1

c = 0

then

R = \tilde{\infty} \left(- e + \lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{10}}{n^{10}}\right)\right)

False

The rate of convergence of the power series

The answer [src]

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\infty

oo

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