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1/((2^(k+1))-1)
Sum of series/ 1/((2^(k+1))-1)

Sum of series 1/((2^(k+1))-1)


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The solution

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

False
The rate of convergence of the power series
Numerical answer [src] 
0.606695152415291763783301523191
0.606695152415291763783301523191
The graph
Sum of series 1/((2^(k+1))-1)

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