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sin(sin(n)/n^(1/3))

  • How to use it?

  • Sum of series:
  • ((-1)^(n+1))/n ((-1)^(n+1))/n
  • (1+1/n)^n (1+1/n)^n
  • sin(sin(n)/n^(1/3)) sin(sin(n)/n^(1/3))
  • 3/(8^(n-1)) 3/(8^(n-1))

  • Identical expressions

  • sin(sin(n)/n^(one / three))
  • sinus of ( sinus of (n) divide by n to the power of (1 divide by 3))
  • sinus of ( sinus of (n) divide by n to the power of (one divide by three))
  • sin(sin(n)/n(1/3))
  • sinsinn/n1/3
  • sinsinn/n^1/3
  • sin(sin(n) divide by n^(1 divide by 3))
Sum of series/ sin(sin(n)/n^(1/3))

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


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The solution

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

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