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  • Sum of series:
  • (2*n-1)/2^n (2*n-1)/2^n
  • (x-1)^n/5^n
  • 1/(4n^2-9) 1/(4n^2-9)
  • sin(n*x)/n^3
  • Identical expressions

  • (x- one)^n/ five ^n
  • (x minus 1) to the power of n divide by 5 to the power of n
  • (x minus one) to the power of n divide by five to the power of n
  • (x-1)n/5n
  • x-1n/5n
  • x-1^n/5^n
  • (x-1)^n divide by 5^n
  • Similar expressions

  • (x+1)^n/5^n

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



=

The solution

You have entered [src]
  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
$$R = 6$$
The answer [src]
/        1   x                         
|      - - + -                         
|        5   5            |  1   x|    
|      -------        for |- - + -| < 1
|       6   x             |  5   5|    
|       - - -                          
|       5   5                          
<                                      
|  oo                                  
| ___                                  
| \  `                                 
|  \    -n         n                   
|  /   5  *(-1 + x)       otherwise    
| /__,                                 
\n = 1                                 
$$\begin{cases} \frac{\frac{x}{5} - \frac{1}{5}}{\frac{6}{5} - \frac{x}{5}} & \text{for}\: \left|{\frac{x}{5} - \frac{1}{5}}\right| < 1 \\\sum_{n=1}^{\infty} 5^{- n} \left(x - 1\right)^{n} & \text{otherwise} \end{cases}$$
Piecewise(((-1/5 + x/5)/(6/5 - x/5), |-1/5 + x/5| < 1), (Sum(5^(-n)*(-1 + x)^n, (n, 1, oo)), True))

    Examples of finding the sum of a series