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Sum of series/ ln(x)/(x^3+x+1)

Sum of series ln(x)/(x^3+x+1)


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The solution

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

False
The answer [src] 
oo*log(x) 
----------
         3
1 + x + x
$$\frac{\infty \log{\left(x \right)}}{x^{3} + x + 1}$$ 
oo*log(x)/(1 + x + x^3)

    Examples of finding the sum of a series

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