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(4*9^n-2^n)/18^n
Sum of series/ (4*9^n-2^n)/18^n

Sum of series (4*9^n-2^n)/18^n


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The solution

You have entered [src] 
  oo           
____           
\   `          
 \       n    n
  \   4*9  - 2 
   )  ---------
  /        n   
 /       18    
/___,          
n = 1
$$\sum_{n=1}^{\infty} \frac{- 2^{n} + 4 \cdot 9^{n}}{18^{n}}$$ 
Sum((4*9^n - 2^n)/18^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{- 2^{n} + 4 \cdot 9^{n}}{18^{n}}$$
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
$$a_{n} = - 2^{n} + 4 \cdot 9^{n}$$
and
$$x_{0} = -18$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-18 + \lim_{n \to \infty} \left|{\frac{2^{n} - 4 \cdot 9^{n}}{2^{n + 1} - 4 \cdot 9^{n + 1}}}\right|\right)$$
Let's take the limit
we find
False

False

$$R = 0$$
The rate of convergence of the power series
The answer [src] 
31/8
$$\frac{31}{8}$$ 
31/8
Numerical answer [src] 
3.87500000000000000000000000000
3.87500000000000000000000000000
The graph
Sum of series (4*9^n-2^n)/18^n

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