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1/(2n+1)*(2n+3)
Sum of series/ 1/(2n+1)*(2n+3)

Sum of series 1/(2n+1)*(2n+3)


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The solution

You have entered [src] 
  oo         
 ___         
 \  `        
  \   2*n + 3
   )  -------
  /   2*n + 1
 /__,        
n = 0
$$\sum_{n=0}^{\infty} \frac{2 n + 3}{2 n + 1}$$ 
Sum((2*n + 3)/(2*n + 1), (n, 0, oo))
The radius of convergence of the power series
Given number:
$$\frac{2 n + 3}{2 n + 1}$$
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
$$a_{n} = \frac{2 n + 3}{2 n + 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(2 n + 3\right)^{2}}{\left(2 n + 1\right) \left(2 n + 5\right)}\right)$$
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/(2n+1)*(2n+3)

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