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((1/n)-arctg(1/n))
Sum of series/ ((1/n)-arctg(1/n))

Sum of series ((1/n)-arctg(1/n))


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The solution

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

False
The rate of convergence of the power series
Numerical answer [src] 
0.275575344433999662718980432286
0.275575344433999662718980432286
The graph
Sum of series ((1/n)-arctg(1/n))

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