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Sum of series/ n^2*x^n

Sum of series n^2*x^n


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The solution

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n = 1
$$\sum_{n=1}^{\infty} n^{2} x^{n}$$ 
Sum(n^2*x^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$n^{2} x^{n}$$
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
$$a_{n} = n^{2}$$
and
$$x_{0} = 0$$
,
$$d = 1$$
,
$$c = 1$$
then
$$R = \lim_{n \to \infty}\left(\frac{n^{2}}{\left(n + 1\right)^{2}}\right)$$
Let's take the limit
we find
$$R = 1$$
The answer [src] 
/x*(-1 - x)              
|----------   for |x| < 1
|        3               
|(-1 + x)                
|                        
|  oo                    
< ___                    
| \  `                   
|  \    2  n             
|  /   n *x    otherwise 
| /__,                   
|n = 1                   
\
$$\begin{cases} \frac{x \left(- x - 1\right)}{\left(x - 1\right)^{3}} & \text{for}\: \left|{x}\right| < 1 \\\sum_{n=1}^{\infty} n^{2} x^{n} & \text{otherwise} \end{cases}$$ 
Piecewise((x*(-1 - x)/(-1 + x)^3, |x| < 1), (Sum(n^2*x^n, (n, 1, oo)), True))

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