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sin(n*pi/2)/n

  • How to use it?

  • Sum of series:
  • 1/7n 1/7n
  • sin(n*pi/2)/n sin(n*pi/2)/n
  • 1/3^n 1/3^n
  • sin^2((n^2+3)/(n^3+2)) sin^2((n^2+3)/(n^3+2))

  • Identical expressions

  • sin(n*pi/ two)/n
  • sinus of (n multiply by Pi divide by 2) divide by n
  • sinus of (n multiply by Pi divide by two) divide by n
  • sin(npi/2)/n
  • sinnpi/2/n
  • sin(n*pi divide by 2) divide by n
Sum of series/ sin(n*pi/2)/n

Sum of series sin(n*pi/2)/n


∑
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The solution

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

False
The rate of convergence of the power series
The answer [src] 
  oo           
____           
\   `          
 \       /pi*n\
  \   sin|----|
   )     \ 2  /
  /   ---------
 /        n    
/___,          
n = 1
$$\sum_{n=1}^{\infty} \frac{\sin{\left(\frac{\pi n}{2} \right)}}{n}$$ 
Sum(sin(pi*n/2)/n, (n, 1, oo))
The graph
Sum of series sin(n*pi/2)/n

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