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  • Sum of series:
  • x*1,2^n
  • (4i+5)/10^2 (4i+5)/10^2
  • x1
  • n/2^n*z^2n

  • Identical expressions

  • sinn/sqrt(x) one /n^ two
  • sinus of n divide by square root of (x)1 divide by n squared
  • sinus of n divide by square root of (x) one divide by n to the power of two
  • sinn/√(x)1/n^2
  • sinn/sqrt(x)1/n2
  • sinn/sqrtx1/n2
  • sinn/sqrt(x)1/n²
  • sinn/sqrt(x)1/n to the power of 2
  • sinn/sqrtx1/n^2
  • sinn divide by sqrt(x)1 divide by n^2
Sum of series/ sinn/sqrt(x)1/n^2

Sum of series sinn/sqrt(x)1/n^2


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The solution

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

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

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