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|1/(n^3+1)|
Sum of series/ |1/(n^3+1)|

Sum of series |1/(n^3+1)|


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The solution

You have entered [src] 
  oo          
____          
\   `         
 \    |  1   |
  \   |------|
  /   | 3    |
 /    |n  + 1|
/___,         
n = 1
$$\sum_{n=1}^{\infty} \left|{\frac{1}{n^{3} + 1}}\right|$$ 
Sum(Abs(1/(n^3 + 1)), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\left|{\frac{1}{n^{3} + 1}}\right|$$
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
$$a_{n} = \frac{1}{\left|{n^{3} + 1}\right|}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{3} + 1}{n^{3} + 1}\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The answer [src] 
  oo        
____        
\   `       
 \      1   
  \   ------
  /        3
 /    1 + n 
/___,       
n = 1
$$\sum_{n=1}^{\infty} \frac{1}{n^{3} + 1}$$ 
Sum(1/(1 + n^3), (n, 1, oo))
The graph
Sum of series |1/(n^3+1)|

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