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(5-((2^n)-1))/(5-(2^n))
Sum of series/ (5-((2^n)-1))/(5-(2^n))

Sum of series (5-((2^n)-1))/(5-(2^n))


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The solution

You have entered [src] 
  oo              
____              
\   `             
 \           n    
  \   5 + - 2  + 1
   )  ------------
  /           n   
 /       5 - 2    
/___,             
n = 6
$$\sum_{n=6}^{\infty} \frac{\left(1 - 2^{n}\right) + 5}{5 - 2^{n}}$$ 
Sum((5 - 2^n + 1)/(5 - 2^n), (n, 6, oo))
The radius of convergence of the power series
Given number:
$$\frac{\left(1 - 2^{n}\right) + 5}{5 - 2^{n}}$$
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
$$a_{n} = \frac{6 - 2^{n}}{5 - 2^{n}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\left(2^{n} - 6\right) \left(2^{n + 1} - 5\right)}{\left(2^{n} - 5\right) \left(2^{n + 1} - 6\right)}}\right|$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The answer [src] 
  oo        
____        
\   `       
 \         n
  \   6 - 2 
   )  ------
  /        n
 /    5 - 2 
/___,       
n = 6
$$\sum_{n=6}^{\infty} \frac{6 - 2^{n}}{5 - 2^{n}}$$ 
Sum((6 - 2^n)/(5 - 2^n), (n, 6, oo))
The graph
Sum of series (5-((2^n)-1))/(5-(2^n))

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