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Sum of series:
(3/4)^n
1/((2n-1)*(2n+1))
1/(n*(n+3))
ln(n+1)/n
Identical expressions
(x^(n- one))/(n(n+ one))
(x to the power of (n minus 1)) divide by (n(n plus 1))
(x to the power of (n minus one)) divide by (n(n plus one))
(x(n-1))/(n(n+1))
xn-1/nn+1
x^n-1/nn+1
(x^(n-1)) divide by (n(n+1))
Similar expressions
(x^(n-1))/(n(n-1))
(x^(n+1))/(n(n+1))

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

The solution

You have entered [src]

  oo           
____           
\   `          
 \       n - 1 
  \     x      
  /   ---------
 /    n*(n + 1)
/___,          
n = 1

$$\sum_{n=1}^{\infty} \frac{x^{n - 1}}{n \left(n + 1\right)}$$

Sum(x^(n - 1)/((n*(n + 1))), (n, 1, oo))

The answer [src]

/1   (2 - 2*x)*log(1 - x)              
|- + --------------------  for |x| <= 1
|x              2                      
|            2*x                       
|                                      
|      oo                              
|    ____                              
<    \   `                             
|     \         n                      
|      \       x                       
|       )  ----------       otherwise  
|      /            2                  
|     /    n*x + x*n                   
|    /___,                             
\    n = 1

$$\begin{cases} \frac{1}{x} + \frac{\left(2 - 2 x\right) \log{\left(1 - x \right)}}{2 x^{2}} & \text{for}\: \left|{x}\right| \leq 1 \\\sum_{n=1}^{\infty} \frac{x^{n}}{n^{2} x + n x} & \text{otherwise} \end{cases}$$

Piecewise((1/x + (2 - 2*x)*log(1 - x)/(2*x^2), |x| <= 1), (Sum(x^n/(n*x + x*n^2), (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

Similar expressions

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

The solution

Examples of finding the sum of a series