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

Sum of series ln(1+1/n)


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The solution

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

False
The rate of convergence of the power series
Numerical answer
The series diverges
The graph
Sum of series ln(1+1/n)

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