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

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


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The solution

You have entered [src] 
  oo           
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\   `          
 \      _______
  \   \/ x + 1 
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 /        x    
/___,          
n = 1
$$\sum_{n=1}^{\infty} \frac{\sqrt{x + 1}}{x}$$ 
Sum(sqrt(x + 1)/x, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\sqrt{x + 1}}{x}$$
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
$$a_{n} = \frac{\sqrt{x + 1}}{x}$$
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*\/ 1 + x 
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     x
$$\frac{\infty \sqrt{x + 1}}{x}$$ 
oo*sqrt(1 + x)/x

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