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sqrt(n+arctgn^2)-(sqrtn)
Sum of series/ sqrt(n+arctgn^2)-(sqrtn)

Sum of series sqrt(n+arctgn^2)-(sqrtn)


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The solution

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

False
The rate of convergence of the power series
The graph
Sum of series sqrt(n+arctgn^2)-(sqrtn)

    Examples of finding the sum of a series

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