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ln((n(n+2))/(n+1)^2)
Sum of series/ ln((n(n+2))/(n+1)^2)

Sum of series ln((n(n+2))/(n+1)^2)


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The solution

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

False
The rate of convergence of the power series
The answer [src] 
  oo                
____                
\   `               
 \       /n*(2 + n)\
  \   log|---------|
  /      |        2|
 /       \ (1 + n) /
/___,               
n = 1
$$\sum_{n=1}^{\infty} \log{\left(\frac{n \left(n + 2\right)}{\left(n + 1\right)^{2}} \right)}$$ 
Sum(log(n*(2 + n)/(1 + n)^2), (n, 1, oo))
The graph
Sum of series ln((n(n+2))/(n+1)^2)

    Examples of finding the sum of a series

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