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Sum of series/ x^n/n^2

Sum of series x^n/n^2


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The solution

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

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