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Sum of series (x+1)^n/5^n



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The solution

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  oo          
____          
\   `         
 \           n
  \   (x + 1) 
   )  --------
  /       n   
 /       5    
/___,         
n = 1         
$$\sum_{n=1}^{\infty} \frac{\left(x + 1\right)^{n}}{5^{n}}$$
Sum((x + 1)^n/5^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\left(x + 1\right)^{n}}{5^{n}}$$
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
$$a_{n} = 5^{- n}$$
and
$$x_{0} = -1$$
,
$$d = 1$$
,
$$c = 1$$
then
$$R = -1 + \lim_{n \to \infty}\left(5^{- n} 5^{n + 1}\right)$$
Let's take the limit
we find
$$R = 4$$
The answer [src]
/      1   x                        
|      - + -                        
|      5   5             |1   x|    
|      -----         for |- + -| < 1
|      4   x             |5   5|    
|      - - -                        
|      5   5                        
<                                   
|  oo                               
| ___                               
| \  `                              
|  \    -n        n                 
|  /   5  *(1 + x)      otherwise   
| /__,                              
\n = 1                              
$$\begin{cases} \frac{\frac{x}{5} + \frac{1}{5}}{\frac{4}{5} - \frac{x}{5}} & \text{for}\: \left|{\frac{x}{5} + \frac{1}{5}}\right| < 1 \\\sum_{n=1}^{\infty} 5^{- n} \left(x + 1\right)^{n} & \text{otherwise} \end{cases}$$
Piecewise(((1/5 + x/5)/(4/5 - x/5), |1/5 + x/5| < 1), (Sum(5^(-n)*(1 + x)^n, (n, 1, oo)), True))

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