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(2/9)^n
Sum of series/ (2/9)^n

Sum of series (2/9)^n


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The solution

You have entered [src] 
  oo      
 ___      
 \  `     
  \      n
  /   2/9 
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n = 1
∑n=1∞(29)n\sum_{n=1}^{\infty} \left(\frac{2}{9}\right)^{n} 
Sum((2/9)^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
(29)n\left(\frac{2}{9}\right)^{n}
It is a series of species
an(cx−x0)dna_{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:
Rd=x0+lim⁡n→∞∣anan+1∣cR^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}
In this case
an=1a_{n} = 1
and
x0=−29x_{0} = - \frac{2}{9}
,
d=1d = 1
,
c=0c = 0
then
False

Let's take the limit
we find
False
The rate of convergence of the power series
1.07.01.52.02.53.03.54.04.55.05.56.06.50.200.30
The answer [src] 
2/7
27\frac{2}{7} 
2/7
Numerical answer [src] 
0.285714285714285714285714285714
0.285714285714285714285714285714
The graph
Sum of series (2/9)^n

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