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



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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))

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