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2-3(0.8)^n
Sum of series/ 2-3(0.8)^n

Sum of series 2-3(0.8)^n


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The solution

You have entered [src] 
  oo              
 ___              
 \  `             
  \   /         n\
  /   \2 - 3*4/5 /
 /__,             
n = 1
∑n=1∞(2−3(45)n)\sum_{n=1}^{\infty} \left(2 - 3 \left(\frac{4}{5}\right)^{n}\right) 
Sum(2 - 3*(4/5)^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
2−3(45)n2 - 3 \left(\frac{4}{5}\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=2−3(45)na_{n} = 2 - 3 \left(\frac{4}{5}\right)^{n}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=lim⁡n→∞∣3(45)n−23(45)n+1−2∣1 = \lim_{n \to \infty} \left|{\frac{3 \left(\frac{4}{5}\right)^{n} - 2}{3 \left(\frac{4}{5}\right)^{n + 1} - 2}}\right|
Let's take the limit
we find
True

False
The rate of convergence of the power series
1.07.01.52.02.53.03.54.04.55.05.56.06.55-5
The answer [src] 
oo
∞\infty 
oo
Numerical answer
The series diverges
The graph
Sum of series 2-3(0.8)^n

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