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(sqrt(n)+1)/(n^3+5)
Sum of series/ (sqrt(n)+1)/(n^3+5)

Sum of series (sqrt(n)+1)/(n^3+5)


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The solution

You have entered [src] 
  oo           
____           
\   `          
 \      ___    
  \   \/ n  + 1
   )  ---------
  /      3     
 /      n  + 5 
/___,          
n = 1
$$\sum_{n=1}^{\infty} \frac{\sqrt{n} + 1}{n^{3} + 5}$$ 
Sum((sqrt(n) + 1)/(n^3 + 5), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\sqrt{n} + 1}{n^{3} + 5}$$
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
$$a_{n} = \frac{\sqrt{n} + 1}{n^{3} + 5}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(\sqrt{n} + 1\right) \left(\left(n + 1\right)^{3} + 5\right)}{\left(n^{3} + 5\right) \left(\sqrt{n + 1} + 1\right)}\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
Numerical answer [src] 
0.739926571791201834271929637053
0.739926571791201834271929637053
The graph
Sum of series (sqrt(n)+1)/(n^3+5)

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