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1\3^n

Sum of series 1\3^n



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The solution

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

$$R = 0$$
The rate of convergence of the power series
The answer [src]
1/2
$$\frac{1}{2}$$
1/2
Numerical answer [src]
0.500000000000000000000000000000
0.500000000000000000000000000000
The graph
Sum of series 1\3^n
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