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1/(n(n+5))
Sum of series/ 1/(n(n+5))

Sum of series 1/(n(n+5))


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The solution

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

False
The rate of convergence of the power series
The answer [src] 
            0                 0  
   -8 + 20*e        -52 + 37*e   
--------------- + ---------------
  /          0\     /          0\
4*\-60 + 60*e /   5*\-60 + 60*e /
$$\frac{-52 + 37 e^{0}}{5 \left(-60 + 60 e^{0}\right)} + \frac{-8 + 20 e^{0}}{4 \left(-60 + 60 e^{0}\right)}$$ 
(-8 + 20*exp_polar(0))/(4*(-60 + 60*exp_polar(0))) + (-52 + 37*exp_polar(0))/(5*(-60 + 60*exp_polar(0)))
Numerical answer [src] 
0.456666666666666666666666666667
0.456666666666666666666666666667
The graph
Sum of series 1/(n(n+5))

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