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1/n^6

  • How to use it?

  • Sum of series:
  • 1/(n(n+2)) 1/(n(n+2))
  • 1/n^6 1/n^6
  • (3^n-4^n)/12^n (3^n-4^n)/12^n
  • 2i 2i

  • Identical expressions

  • one /n^ six
  • 1 divide by n to the power of 6
  • one divide by n to the power of six
  • 1/n6
  • 1/n⁶
  • 1 divide by n^6
Sum of series/ 1/n^6

Sum of series 1/n^6


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The solution

You have entered [src] 
  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
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{6}}{n^{6}}\right)$$
Let's take the limit
we find
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
Sum of series 1/n^6

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