Sum of series 1/(nlnn)

The solution

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  oo          
 ___          
 \  `         
  \      1    
   )  --------
  /   n*log(n)
 /__,         
n = 1

$$\sum_{n=1}^{\infty} \frac{1}{n \log{\left(n \right)}}$$

Sum(1/(n*log(n)), (n, 1, oo))

The radius of convergence of the power series

Given number:
$$\frac{1}{n \log{\left(n \right)}}$$
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
$$a_{n} = \frac{1}{n \log{\left(n \right)}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right) \log{\left(n + 1 \right)} \left|{\frac{1}{\log{\left(n \right)}}}\right|}{n}\right)$$
Let's take the limit
we find

True

False

The rate of convergence of the power series

Numerical answer [src]

Sum(1/(n*log(n)), (n, 1, oo))

Sum(1/(n*log(n)), (n, 1, oo))

The graph

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```
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```
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```
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```
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Sum of series 1/(nlnn)

The solution

Examples of finding the sum of a series