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Sum of series (x^n)/(n*(n+1))



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The solution

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  oo           
____           
\   `          
 \         n   
  \       x    
  /   ---------
 /    n*(n + 1)
/___,          
n = 1          
$$\sum_{n=1}^{\infty} \frac{x^{n}}{n \left(n + 1\right)}$$
Sum(x^n/((n*(n + 1))), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{x^{n}}{n \left(n + 1\right)}$$
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
$$a_{n} = \frac{1}{n \left(n + 1\right)}$$
and
$$x_{0} = 0$$
,
$$d = 1$$
,
$$c = 1$$
then
$$R = \lim_{n \to \infty}\left(\frac{n + 2}{n}\right)$$
Let's take the limit
we find
$$R = 1$$
The answer [src]
/  /2   (2 - 2*x)*log(1 - x)\              
|x*|- + --------------------|              
|  |x             2         |              
|  \             x          /              
|----------------------------  for |x| <= 1
|             2                            
|                                          
|          oo                              
<        ____                              
|        \   `                             
|         \       n                        
|          \     x                         
|           )  ------           otherwise  
|          /        2                      
|         /    n + n                       
|        /___,                             
\        n = 1                             
$$\begin{cases} \frac{x \left(\frac{2}{x} + \frac{\left(2 - 2 x\right) \log{\left(1 - x \right)}}{x^{2}}\right)}{2} & \text{for}\: \left|{x}\right| \leq 1 \\\sum_{n=1}^{\infty} \frac{x^{n}}{n^{2} + n} & \text{otherwise} \end{cases}$$
Piecewise((x*(2/x + (2 - 2*x)*log(1 - x)/x^2)/2, |x| <= 1), (Sum(x^n/(n + n^2), (n, 1, oo)), True))

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