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Sum of series/ 1/n^6

Sum of series 1/n^6

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  oo    
____    
\   `   
 \    1 
  \   --
  /    6
 /    n 
/___,   
n = 1

\sum_{n=1}^{\infty} \frac{1}{n^{6}}

Sum(1/(n^6), (n, 1, oo))

The radius of convergence of the power series

Given number:

\frac{1}{n^{6}}

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^{6}}

and

x_{0} = 0

d = 0

c = 1

then

1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{6}}{n^{6}}\right)

True

False

The rate of convergence of the power series

The answer [src]

  6
pi 
---
945

\frac{\pi^{6}}{945}

pi^6/945

Numerical answer [src]

1.01734306198444913971451792979

1.01734306198444913971451792979

The graph