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1/n(sqrt(n+1)-sqrt(n-1))
Sum of series/ 1/n(sqrt(n+1)-sqrt(n-1))

Sum of series 1/n(sqrt(n+1)-sqrt(n-1))


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The solution

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

False
The rate of convergence of the power series
Numerical answer [src] 
3.04403454948904226007475434988
3.04403454948904226007475434988
The graph
Sum of series 1/n(sqrt(n+1)-sqrt(n-1))

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