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1^(n-1)/8^n
Sum of series/ 1^(n-1)/8^n

Sum of series 1^(n-1)/8^n


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The solution

You have entered [src] 
  oo        
____        
\   `       
 \     n - 1
  \   1     
   )  ------
  /      n  
 /      8   
/___,       
n = 1
$$\sum_{n=1}^{\infty} \frac{1^{n - 1}}{8^{n}}$$ 
Sum(1^(n - 1)/8^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{1^{n - 1}}{8^{n}}$$
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
$$a_{n} = 1$$
and
$$x_{0} = -8$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-8 + \lim_{n \to \infty} 1\right)$$
Let's take the limit
we find
False

$$R = 0$$
The rate of convergence of the power series
The answer [src] 
1/7
$$\frac{1}{7}$$ 
1/7
Numerical answer [src] 
0.142857142857142857142857142857
0.142857142857142857142857142857
The graph
Sum of series 1^(n-1)/8^n

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