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(-1/2)^n
Sum of series/ (-1/2)^n

Sum of series (-1/2)^n


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The solution

You have entered [src] 
  oo       
 ___       
 \  `      
  \       n
  /   -1/2 
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n = 0
∑n=0∞(−12)n\sum_{n=0}^{\infty} \left(- \frac{1}{2}\right)^{n} 
Sum((-1/2)^n, (n, 0, oo))
The radius of convergence of the power series
Given number:
(−12)n\left(- \frac{1}{2}\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=12x_{0} = \frac{1}{2}
,
d=1d = 1
,
c=0c = 0
then
False

Let's take the limit
we find
False
The rate of convergence of the power series
0.06.00.51.01.52.02.53.03.54.04.55.05.50.01.5
The answer [src] 
2/3
23\frac{2}{3} 
2/3
Numerical answer [src] 
0.666666666666666666666666666667
0.666666666666666666666666666667
The graph
Sum of series (-1/2)^n

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