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Sum of series/ 1/((x(x+1)))^(1/2)

Sum of series 1/((x(x+1)))^(1/2)


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The solution

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

False
The answer [src] 
      oo     
-------------
  ___________
\/ x*(1 + x)
$$\frac{\infty}{\sqrt{x \left(x + 1\right)}}$$ 
oo/sqrt(x*(1 + x))

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