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((5n+1)-(5n-4))/((5n-4)(5n+1))
Sum of series/ ((5n+1)-(5n-4))/((5n-4)(5n+1))

Sum of series ((5n+1)-(5n-4))/((5n-4)(5n+1))


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The solution

You have entered [src] 
  oo                     
 ___                     
 \  `                    
  \    5*n + 1 + -5*n + 4
   )  -------------------
  /   (5*n - 4)*(5*n + 1)
 /__,                    
n = 1
$$\sum_{n=1}^{\infty} \frac{\left(4 - 5 n\right) + \left(5 n + 1\right)}{\left(5 n - 4\right) \left(5 n + 1\right)}$$ 
Sum((5*n + 1 - 5*n + 4)/(((5*n - 4)*(5*n + 1))), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\left(4 - 5 n\right) + \left(5 n + 1\right)}{\left(5 n - 4\right) \left(5 n + 1\right)}$$
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
$$a_{n} = \frac{5}{\left(5 n - 4\right) \left(5 n + 1\right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\left(5 n + 6\right) \left|{\frac{1}{5 n - 4}}\right|\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The answer [src] 
5*Gamma(11/5)
-------------
 6*Gamma(6/5)
$$\frac{5 \Gamma\left(\frac{11}{5}\right)}{6 \Gamma\left(\frac{6}{5}\right)}$$ 
5*gamma(11/5)/(6*gamma(6/5))
Numerical answer [src] 
1.00000000000000000000000000000
1.00000000000000000000000000000
The graph
Sum of series ((5n+1)-(5n-4))/((5n-4)(5n+1))

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