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Sum of series:
1/(3n+2)(3n+1)
(x-1)^n/5^n
1/4^x
(2*n-1)/2^n
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

The Sum of the Power Series
```
x^n/n
```
```
(x-1)^n
```
Factorial
```
1/2^(n!)
```
```
n^2/n!
```
```
x^n/n!
```
```
k!/(n!*(n+k)!)
```
Flint Hills Series
```
csc(n)^2/n^3
```
Basel problem
```
1/n^2
```
```
1/n^4
```
```
1/n^6
```
Harmonic series
```
1/n
```
Grandi's series
```
(-1)^n
```
Alternating series
```
(-1)^(n + 1)/n
```
```
(n + 2)*(-1)^(n - 1)
```
```
(3*n - 1)/(-5)^n
```
```
(-1)^(n - 1)*n/(6*n - 5)
```
Newton–Mercator series
```
(-1)^(n + 1)/n*x^n
```
Exam the series for convergence
```
(3*n - 1)/(-5)^n
```

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How to use it?

Identical expressions

Similar expressions

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

The solution

Examples of finding the sum of a series