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(3^n+1)/6^n
Sum of series/ (3^n+1)/6^n

Sum of series (3^n+1)/6^n


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The solution

You have entered [src] 
  oo        
____        
\   `       
 \     n    
  \   3  + 1
   )  ------
  /      n  
 /      6   
/___,       
n = 1
$$\sum_{n=1}^{\infty} \frac{3^{n} + 1}{6^{n}}$$ 
Sum((3^n + 1)/6^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{3^{n} + 1}{6^{n}}$$
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
$$a_{n} = 3^{n} + 1$$
and
$$x_{0} = -6$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-6 + \lim_{n \to \infty}\left(\frac{3^{n} + 1}{3^{n + 1} + 1}\right)\right)$$
Let's take the limit
we find
False

False

$$R = 0$$
The rate of convergence of the power series
The answer [src] 
6/5
$$\frac{6}{5}$$ 
6/5
Numerical answer [src] 
1.20000000000000000000000000000
1.20000000000000000000000000000
The graph
Sum of series (3^n+1)/6^n

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