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

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


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The solution

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

False
The rate of convergence of the power series
The answer [src] 
  oo             
____             
\   `            
 \        /  1  \
  \   atan|-----|
  /       |3 ___|
 /        \\/ n /
/___,            
n = 1
$$\sum_{n=1}^{\infty} \operatorname{atan}{\left(\frac{1}{\sqrt[3]{n}} \right)}$$ 
Sum(atan(n^(-1/3)), (n, 1, oo))
Numerical answer
The series diverges
The graph
Sum of series arctg(1/n^(1/3))

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