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arcctg((n+1)/(n^2-3))
Sum of series/ arcctg((n+1)/(n^2-3))

Sum of series arcctg((n+1)/(n^2-3))


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The solution

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  oo              
____              
\   `             
 \        /n + 1 \
  \   acot|------|
  /       | 2    |
 /        \n  - 3/
/___,             
n = 1
$$\sum_{n=1}^{\infty} \operatorname{acot}{\left(\frac{n + 1}{n^{2} - 3} \right)}$$ 
Sum(acot((n + 1)/(n^2 - 3)), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\operatorname{acot}{\left(\frac{n + 1}{n^{2} - 3} \right)}$$
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
$$a_{n} = \operatorname{acot}{\left(\frac{n + 1}{n^{2} - 3} \right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\operatorname{acot}{\left(\frac{n + 1}{n^{2} - 3} \right)}}{\operatorname{acot}{\left(\frac{n + 2}{\left(n + 1\right)^{2} - 3} \right)}}}\right|$$
Let's take the limit
we find
False

False
The rate of convergence of the power series
The graph
Sum of series arcctg((n+1)/(n^2-3))

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