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Sum of series 1/3^i



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The solution

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  oo     
 ___     
 \  `    
  \    -I
  /   3  
 /__,    
n = 0    
$$\sum_{n=0}^{\infty} \left(\frac{1}{3}\right)^{i}$$
Sum((1/3)^i, (n, 0, oo))
The radius of convergence of the power series
Given number:
$$\left(\frac{1}{3}\right)^{i}$$
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
$$a_{n} = \left(\frac{1}{3}\right)^{i}$$
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]
    -I
oo*3  
$$\infty 3^{- i}$$
oo*3^(-i)
Numerical answer
The series diverges

    Examples of finding the sum of a series