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Sum of series x^(n)



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The solution

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