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(n^3+n+5)/(n+2)
Sum of series/ (n^3+n+5)/(n+2)

Sum of series (n^3+n+5)/(n+2)


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The solution

You have entered [src] 
  oo            
____            
\   `           
 \     3        
  \   n  + n + 5
  /   ----------
 /      n + 2   
/___,           
n = 1
$$\sum_{n=1}^{\infty} \frac{\left(n^{3} + n\right) + 5}{n + 2}$$ 
Sum((n^3 + n + 5)/(n + 2), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\left(n^{3} + n\right) + 5}{n + 2}$$
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
$$a_{n} = \frac{n^{3} + n + 5}{n + 2}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 3\right) \left(n^{3} + n + 5\right)}{\left(n + 2\right) \left(n + \left(n + 1\right)^{3} + 6\right)}\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The answer [src] 
oo
$$\infty$$ 
oo
Numerical answer
The series diverges
The graph
Sum of series (n^3+n+5)/(n+2)

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