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1/(n*(lnn+1)^2)

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  • Sum of series:
  • x^n/n
  • 1/n! 1/n!
  • (9835^(2k/9835))/(9835+9835^(2k/9835)) (9835^(2k/9835))/(9835+9835^(2k/9835))
  • nx^n!

  • Identical expressions

  • one /(n*(lnn+ one)^ two)
  • 1 divide by (n multiply by (lnn plus 1) squared )
  • one divide by (n multiply by (lnn plus one) to the power of two)
  • 1/(n*(lnn+1)2)
  • 1/n*lnn+12
  • 1/(n*(lnn+1)²)
  • 1/(n*(lnn+1) to the power of 2)
  • 1/(n(lnn+1)^2)
  • 1/(n(lnn+1)2)
  • 1/nlnn+12
  • 1/nlnn+1^2
  • 1 divide by (n*(lnn+1)^2)

  • Similar expressions

  • 1/(n*(lnn-1)^2)
Sum of series/ 1/(n*(lnn+1)^2)

Sum of series 1/(n*(lnn+1)^2)


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The solution

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

False
The rate of convergence of the power series
The graph
Sum of series 1/(n*(lnn+1)^2)

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