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  • Sum of series:
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  • arctan(n)/(n!)^2 arctan(n)/(n!)^2
  • arctg(5/n)/n! arctg(5/n)/n!
  • ln((n^2+n+3)/(n^2+n+2)) ln((n^2+n+3)/(n^2+n+2))

  • Identical expressions

  • sinx/(n(n)^(one / three))
  • sinus of x divide by (n(n) to the power of (1 divide by 3))
  • sinus of x divide by (n(n) to the power of (one divide by three))
  • sinx/(n(n)(1/3))
  • sinx/nn1/3
  • sinx/nn^1/3
  • sinx divide by (n(n)^(1 divide by 3))
Sum of series/ sinx/(n(n)^(1/3))

Sum of series sinx/(n(n)^(1/3))


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The solution

You have entered [src] 
  oo         
____         
\   `        
 \     sin(x)
  \   -------
  /     3 ___
 /    n*\/ n 
/___,        
n = 1
$$\sum_{n=1}^{\infty} \frac{\sin{\left(x \right)}}{\sqrt[3]{n} n}$$ 
Sum(sin(x)/((n*n^(1/3))), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\sin{\left(x \right)}}{\sqrt[3]{n} n}$$
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
$$a_{n} = \frac{\sin{\left(x \right)}}{n^{\frac{4}{3}}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{\frac{4}{3}}}{n^{\frac{4}{3}}}\right)$$
Let's take the limit
we find
True

False
The answer [src] 
  oo        
____        
\   `       
 \    sin(x)
  \   ------
  /     4/3 
 /     n    
/___,       
n = 1
$$\sum_{n=1}^{\infty} \frac{\sin{\left(x \right)}}{n^{\frac{4}{3}}}$$ 
Sum(sin(x)/n^(4/3), (n, 1, oo))

    Examples of finding the sum of a series

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