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  • Sum of series:
  • 1/((4n-3)(4n+1)) 1/((4n-3)(4n+1))
  • (-1)^n*x^n/(2*n+1)
  • (arcsin(1/n))^n (arcsin(1/n))^n
  • sen(ix)

  • Identical expressions

  • (- one)^n*x^n/(two *n+ one)
  • ( minus 1) to the power of n multiply by x to the power of n divide by (2 multiply by n plus 1)
  • ( minus one) to the power of n multiply by x to the power of n divide by (two multiply by n plus one)
  • (-1)n*xn/(2*n+1)
  • -1n*xn/2*n+1
  • (-1)^nx^n/(2n+1)
  • (-1)nxn/(2n+1)
  • -1nxn/2n+1
  • -1^nx^n/2n+1
  • (-1)^n*x^n divide by (2*n+1)

  • Similar expressions

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

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


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The solution

You have entered [src] 
  oo          
____          
\   `         
 \        n  n
  \   (-1) *x 
  /   --------
 /    2*n + 1 
/___,         
n = 1
$$\sum_{n=1}^{\infty} \frac{\left(-1\right)^{n} x^{n}}{2 n + 1}$$ 
Sum(((-1)^n*x^n)/(2*n + 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\left(-1\right)^{n} x^{n}}{2 n + 1}$$
It is a series of species
$$a_{n} \left(c x - x_{0}\right)^{d n}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}$$
In this case
$$a_{n} = \frac{1}{2 n + 1}$$
and
$$x_{0} = 0$$
,
$$d = 1$$
,
$$c = -1$$
then
$$R = - \lim_{n \to \infty}\left(\frac{2 n + 3}{2 n + 1}\right)$$
Let's take the limit
we find
$$R = -1$$
The answer [src] 
/   /          /  ___\\                          
|   |3   3*atan\\/ x /|                          
|-x*|- - -------------|                          
|   |x         3/2    |                          
|   \         x       /                          
|-----------------------  for And(x <= 1, x > -1)
|           3                                    
|                                                
<      oo                                        
|    ____                                        
|    \   `                                       
|     \        n  n                              
|      \   (-1) *x                               
|      /   --------              otherwise       
|     /    1 + 2*n                               
|    /___,                                       
\    n = 1
$$\begin{cases} - \frac{x \left(\frac{3}{x} - \frac{3 \operatorname{atan}{\left(\sqrt{x} \right)}}{x^{\frac{3}{2}}}\right)}{3} & \text{for}\: x \leq 1 \wedge x > -1 \\\sum_{n=1}^{\infty} \frac{\left(-1\right)^{n} x^{n}}{2 n + 1} & \text{otherwise} \end{cases}$$ 
Piecewise((-x*(3/x - 3*atan(sqrt(x))/x^(3/2))/3, (x <= 1)∧(x > -1)), (Sum((-1)^n*x^n/(1 + 2*n), (n, 1, oo)), True))

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