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9/(n^2+9n+18)
Sum of series/ 9/(n^2+9n+18)

Sum of series 9/(n^2+9n+18)


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The solution

You have entered [src] 
  oo               
____               
\   `              
 \          9      
  \   -------------
  /    2           
 /    n  + 9*n + 18
/___,              
n = 1
$$\sum_{n=1}^{\infty} \frac{9}{\left(n^{2} + 9 n\right) + 18}$$ 
Sum(9/(n^2 + 9*n + 18), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{9}{\left(n^{2} + 9 n\right) + 18}$$
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
$$a_{n} = \frac{9}{n^{2} + 9 n + 18}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{9 \left(n + \frac{\left(n + 1\right)^{2}}{9} + 3\right)}{n^{2} + 9 n + 18}\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The answer [src] 
  /          0\      /        0\ 
3*\-13 + 10*e /    9*\-2 + 3*e / 
--------------- + ---------------
  /          0\     /          0\
4*\-15 + 15*e /   4*\-15 + 15*e /
$$\frac{3 \left(-13 + 10 e^{0}\right)}{4 \left(-15 + 15 e^{0}\right)} + \frac{9 \left(-2 + 3 e^{0}\right)}{4 \left(-15 + 15 e^{0}\right)}$$ 
3*(-13 + 10*exp_polar(0))/(4*(-15 + 15*exp_polar(0))) + 9*(-2 + 3*exp_polar(0))/(4*(-15 + 15*exp_polar(0)))
Numerical answer [src] 
1.85000000000000000000000000000
1.85000000000000000000000000000
The graph
Sum of series 9/(n^2+9n+18)

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