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Sum of series (x-1)^n/3^n



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The solution

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  oo          
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 \           n
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  /       n   
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n = 1         
$$\sum_{n=1}^{\infty} \frac{\left(x - 1\right)^{n}}{3^{n}}$$
Sum((x - 1)^n/3^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\left(x - 1\right)^{n}}{3^{n}}$$
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
$$a_{n} = 3^{- n}$$
and
$$x_{0} = 1$$
,
$$d = 1$$
,
$$c = 1$$
then
$$R = 1 + \lim_{n \to \infty}\left(3^{- n} 3^{n + 1}\right)$$
Let's take the limit
we find
$$R = 4$$
The answer [src]
/        1   x                         
|      - - + -                         
|        3   3            |  1   x|    
|      -------        for |- - + -| < 1
|       4   x             |  3   3|    
|       - - -                          
|       3   3                          
<                                      
|  oo                                  
| ___                                  
| \  `                                 
|  \    -n         n                   
|  /   3  *(-1 + x)       otherwise    
| /__,                                 
\n = 1                                 
$$\begin{cases} \frac{\frac{x}{3} - \frac{1}{3}}{\frac{4}{3} - \frac{x}{3}} & \text{for}\: \left|{\frac{x}{3} - \frac{1}{3}}\right| < 1 \\\sum_{n=1}^{\infty} 3^{- n} \left(x - 1\right)^{n} & \text{otherwise} \end{cases}$$
Piecewise(((-1/3 + x/3)/(4/3 - x/3), |-1/3 + x/3| < 1), (Sum(3^(-n)*(-1 + x)^n, (n, 1, oo)), True))

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