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  • Sum of series:
  • 1/((3n-2)(3n+1)) 1/((3n-2)(3n+1))
  • 3/(n(n+2)) 3/(n(n+2))
  • 6/9n^2+12n-5 6/9n^2+12n-5
  • 3i

  • Identical expressions

  • (-cos(n*(pi))/(n*(pi)))*(sen((n*(pi)*x))/ two)
  • ( minus co sinus of e of (n multiply by ( Pi )) divide by (n multiply by ( Pi ))) multiply by (sen((n multiply by ( Pi ) multiply by x)) divide by 2)
  • ( minus co sinus of e of (n multiply by ( Pi )) divide by (n multiply by ( Pi ))) multiply by (sen((n multiply by ( Pi ) multiply by x)) divide by two)
  • (-cos(n(pi))/(n(pi)))(sen((n(pi)x))/2)
  • -cosnpi/npisennpix/2
  • (-cos(n*(pi)) divide by (n*(pi)))*(sen((n*(pi)*x)) divide by 2)

  • Similar expressions

  • (cos(n*(pi))/(n*(pi)))*(sen((n*(pi)*x))/2)
Sum of series/ (-cos(n*(pi))/(n*(pi)))*(sen((n*(pi)*x))/2)

Sum of series (-cos(n*(pi))/(n*(pi)))*(sen((n*(pi)*x))/2)


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The solution

You have entered [src] 
  oo                         
 ___                         
 \  `                        
  \   -cos(n*pi)  sin(n*pi*x)
   )  -----------*-----------
  /       n*pi         2     
 /__,                        
n = 1
$$\sum_{n=1}^{\infty} \frac{\sin{\left(x \pi n \right)}}{2} \frac{\left(-1\right) \cos{\left(\pi n \right)}}{\pi n}$$ 
Sum(((-cos(n*pi))/((n*pi)))*(sin((n*pi)*x)/2), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\sin{\left(x \pi n \right)}}{2} \frac{\left(-1\right) \cos{\left(\pi n \right)}}{\pi n}$$
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
$$a_{n} = - \frac{\sin{\left(\pi n x \right)} \cos{\left(\pi n \right)}}{2 \pi n}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right) \left|{\frac{\sin{\left(\pi n x \right)} \cos{\left(\pi n \right)}}{\sin{\left(\pi x \left(n + 1\right) \right)} \cos{\left(\pi \left(n + 1\right) \right)}}}\right|}{n}\right)$$
Let's take the limit
we find
True

False
The answer [src] 
  oo                         
 ___                         
 \  `                        
  \   -cos(pi*n)*sin(pi*n*x) 
   )  -----------------------
  /            2*pi*n        
 /__,                        
n = 1
$$\sum_{n=1}^{\infty} - \frac{\sin{\left(\pi n x \right)} \cos{\left(\pi n \right)}}{2 \pi n}$$ 
Sum(-cos(pi*n)*sin(pi*n*x)/(2*pi*n), (n, 1, oo))

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