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2/(n^2+6n+8)
Sum of series/ 2/(n^2+6n+8)

Sum of series 2/(n^2+6n+8)


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The solution

You have entered [src] 
  oo              
____              
\   `             
 \         2      
  \   ------------
  /    2          
 /    n  + 6*n + 8
/___,             
n = 1
$$\sum_{n=1}^{\infty} \frac{2}{\left(n^{2} + 6 n\right) + 8}$$ 
Sum(2/(n^2 + 6*n + 8), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{2}{\left(n^{2} + 6 n\right) + 8}$$
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
$$a_{n} = \frac{2}{n^{2} + 6 n + 8}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{2 \left(3 n + \frac{\left(n + 1\right)^{2}}{2} + 7\right)}{n^{2} + 6 n + 8}\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The answer [src] 
7/12
$$\frac{7}{12}$$ 
7/12
Numerical answer [src] 
0.583333333333333333333333333333
0.583333333333333333333333333333
The graph
Sum of series 2/(n^2+6n+8)

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