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

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


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The solution

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  oo             
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\   `            
 \       /sin(n)\
  \   sin|------|
  /      |3 ___ |
 /       \\/ n  /
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n = 1
∑n=1∞sin⁡(sin⁡(n)n3)\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⁡(sin⁡(n)n3)\sin{\left(\frac{\sin{\left(n \right)}}{\sqrt[3]{n}} \right)}
It is a series of species
an(cx−x0)dna_{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:
Rd=x0+lim⁡n→∞∣anan+1∣cR^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}
In this case
an=sin⁡(sin⁡(n)n3)a_{n} = \sin{\left(\frac{\sin{\left(n \right)}}{\sqrt[3]{n}} \right)}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=lim⁡n→∞∣sin⁡(sin⁡(n)n3)sin⁡(sin⁡(n+1)n+13)∣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→∞∣sin⁡(sin⁡(n)n3)sin⁡(sin⁡(n+1)n+13)∣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
1.07.01.52.02.53.03.54.04.55.05.56.06.502
The graph
Sum of series sin(sin(n)/n^(1/3))

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