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ln(k+1/k+2)
Sum of series/ ln(k+1/k+2)

Sum of series ln(k+1/k+2)


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The solution

You have entered [src] 
  oo                 
 ___                 
 \  `                
  \       /    1    \
   )   log|k + - + 2|
  /       \    k    /
 /__,                
k = 20
$$\sum_{k=20}^{\infty} \log{\left(\left(k + \frac{1}{k}\right) + 2 \right)}$$ 
Sum(log(k + 1/k + 2), (k, 20, oo))
The radius of convergence of the power series
Given number:
$$\log{\left(\left(k + \frac{1}{k}\right) + 2 \right)}$$
It is a series of species
$$a_{k} \left(c x - x_{0}\right)^{d k}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{k \to \infty} \left|{\frac{a_{k}}{a_{k + 1}}}\right|}{c}$$
In this case
$$a_{k} = \log{\left(k + 2 + \frac{1}{k} \right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{k \to \infty}\left(\frac{\log{\left(k + 2 + \frac{1}{k} \right)}}{\log{\left(k + 3 + \frac{1}{k + 1} \right)}}\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The answer [src] 
  oo                 
 ___                 
 \  `                
  \       /        1\
   )   log|2 + k + -|
  /       \        k/
 /__,                
k = 20
$$\sum_{k=20}^{\infty} \log{\left(k + 2 + \frac{1}{k} \right)}$$ 
Sum(log(2 + k + 1/k), (k, 20, oo))
The graph
Sum of series ln(k+1/k+2)

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