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log(1+1/n)/n
Sum of series/ log(1+1/n)/n

Sum of series log(1+1/n)/n


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The solution

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

False
The rate of convergence of the power series
Numerical answer [src] 
0.564599706384424320592667709038
0.564599706384424320592667709038
The graph
Sum of series log(1+1/n)/n

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