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1/((n+6)*(n+7))
Sum of series/ 1/((n+6)*(n+7))

Sum of series 1/((n+6)*(n+7))


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The solution

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

False
The rate of convergence of the power series
The answer [src] 
1/7
$$\frac{1}{7}$$ 
1/7
Numerical answer [src] 
0.142857142857142857142857142857
0.142857142857142857142857142857
The graph
Sum of series 1/((n+6)*(n+7))

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