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1/(4*n-3)(4*n+1)
Sum of series/ 1/(4*n-3)(4*n+1)

Sum of series 1/(4*n-3)(4*n+1)


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The solution

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  oo         
 ___         
 \  `        
  \   4*n + 1
   )  -------
  /   4*n - 3
 /__,        
n = 1
$$\sum_{n=1}^{\infty} \frac{4 n + 1}{4 n - 3}$$ 
Sum((4*n + 1)/(4*n - 3), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{4 n + 1}{4 n - 3}$$
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
$$a_{n} = \frac{4 n + 1}{4 n - 3}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(4 n + 1\right)^{2} \left|{\frac{1}{4 n - 3}}\right|}{4 n + 5}\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The answer [src] 
oo
$$\infty$$ 
oo
Numerical answer
The series diverges
The graph
Sum of series 1/(4*n-3)(4*n+1)

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