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

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


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The solution

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

False
The answer [src] 
     oo    
-----------
      1 + x
-1 + 2
$$\frac{\infty}{2^{x + 1} - 1}$$ 
oo/(-1 + 2^(1 + x))

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