Sum of series ln(x)^n

The solution

You have entered [src]

  oo         
 ___         
 \  `        
  \      n   
  /   log (x)
 /__,        
n = 1

$$\sum_{n=1}^{\infty} \log{\left(x \right)}^{n}$$

Sum(log(x)^n, (n, 1, oo))

The answer [src]

/   log(x)                      
| ----------    for |log(x)| < 1
| 1 - log(x)                    
|                               
|  oo                           
< ___                           
| \  `                          
|  \      n                     
|  /   log (x)     otherwise    
| /__,                          
\n = 1

$$\begin{cases} \frac{\log{\left(x \right)}}{1 - \log{\left(x \right)}} & \text{for}\: \left|{\log{\left(x \right)}}\right| < 1 \\\sum_{n=1}^{\infty} \log{\left(x \right)}^{n} & \text{otherwise} \end{cases}$$

Piecewise((log(x)/(1 - log(x)), Abs(log(x)) < 1), (Sum(log(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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Identical expressions

Sum of series ln(x)^n

The solution

Examples of finding the sum of a series