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Sum of series:
1/((2n-1)(2n+1))
1/3
6/(3n-1)(3n+5)
ln(n)/n^2
Identical expressions
(x^(2n+ one))/(2n+ one)
(x to the power of (2n plus 1)) divide by (2n plus 1)
(x to the power of (2n plus one)) divide by (2n plus one)
(x(2n+1))/(2n+1)
x2n+1/2n+1
x^2n+1/2n+1
(x^(2n+1)) divide by (2n+1)
Similar expressions
(x^(2n-1))/(2n+1)
(x^(2n+1))/(2n-1)

Sum of series (x^(2n+1))/(2n+1)

The solution

You have entered [src]

  oo          
____          
\   `         
 \     2*n + 1
  \   x       
  /   --------
 /    2*n + 1 
/___,         
n = 1

$$\sum_{n=1}^{\infty} \frac{x^{2 n + 1}}{2 n + 1}$$

Sum(x^(2*n + 1)/(2*n + 1), (n, 1, oo))

The answer [src]

  // 2 /  3    3*atanh(x)\                        \
  ||x *|- -- + ----------|                        |
  ||   |   2        3    |                        |
  ||   \  x        x     /                        |
  ||----------------------  for And(x > -1, x < 1)|
  ||          3                                   |
  ||                                              |
  ||      oo                                      |
x*|<    ____                                      |
  ||    \   `                                     |
  ||     \       2*n                              |
  ||      \     x                                 |
  ||      /   -------             otherwise       |
  ||     /    1 + 2*n                             |
  ||    /___,                                     |
  ||    n = 1                                     |
  \\                                              /

$$x \left(\begin{cases} \frac{x^{2} \left(- \frac{3}{x^{2}} + \frac{3 \operatorname{atanh}{\left(x \right)}}{x^{3}}\right)}{3} & \text{for}\: x > -1 \wedge x < 1 \\\sum_{n=1}^{\infty} \frac{x^{2 n}}{2 n + 1} & \text{otherwise} \end{cases}\right)$$

x*Piecewise((x^2*(-3/x^2 + 3*atanh(x)/x^3)/3, (x > -1)∧(x < 1)), (Sum(x^(2*n)/(1 + 2*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

Similar expressions

Sum of series (x^(2n+1))/(2n+1)

The solution

Examples of finding the sum of a series