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ln(n)/(n*ln(ln(n)))*(-1)^n
Sum of series/ ln(n)/(n*ln(ln(n)))*(-1)^n

Sum of series ln(n)/(n*ln(ln(n)))*(-1)^n


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The solution

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  oo                      
 ___                      
 \  `                     
  \        log(n)        n
   )   -------------*(-1) 
  /    n*log(log(n))      
 /__,                     
n = 10
$$\sum_{n=10}^{\infty} \left(-1\right)^{n} \frac{\log{\left(n \right)}}{n \log{\left(\log{\left(n \right)} \right)}}$$ 
Sum((log(n)/((n*log(log(n)))))*(-1)^n, (n, 10, oo))
The radius of convergence of the power series
Given number:
$$\left(-1\right)^{n} \frac{\log{\left(n \right)}}{n \log{\left(\log{\left(n \right)} \right)}}$$
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
$$a_{n} = \frac{\log{\left(n \right)}}{n \log{\left(\log{\left(n \right)} \right)}}$$
and
$$x_{0} = 1$$
,
$$d = 1$$
,
$$c = 0$$
then
$$R = \tilde{\infty} \left(1 + \lim_{n \to \infty}\left(\frac{\left(n + 1\right) \left|{\frac{\log{\left(n \right)} \log{\left(\log{\left(n + 1 \right)} \right)}}{\log{\left(\log{\left(n \right)} \right)}}}\right|}{n \log{\left(n + 1 \right)}}\right)\right)$$
Let's take the limit
we find
False
The rate of convergence of the power series
The answer [src] 
  oo                
____                
\   `               
 \          n       
  \     (-1) *log(n)
  /    -------------
 /     n*log(log(n))
/___,               
n = 10
$$\sum_{n=10}^{\infty} \frac{\left(-1\right)^{n} \log{\left(n \right)}}{n \log{\left(\log{\left(n \right)} \right)}}$$ 
Sum((-1)^n*log(n)/(n*log(log(n))), (n, 10, oo))
The graph
Sum of series ln(n)/(n*ln(ln(n)))*(-1)^n

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