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Sum of series:
x^(n+1)/(n(n+1))
1/4n^2-1
cos(2n)/n^2
((8)(ln(x)))/(x^3)
Identical expressions
x^(n+ one)/(n(n+ one))
x to the power of (n plus 1) divide by (n(n plus 1))
x to the power of (n plus 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]

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

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

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