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(7^(n+1))(8^-n)
Sum of series/ (7^(n+1))(8^-n)

Sum of series (7^(n+1))(8^-n)


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The solution

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  oo            
 ___            
 \  `           
  \    n + 1  -n
  /   7     *8  
 /__,           
n = 1
∑n=1∞7n+1⋅8−n\sum_{n=1}^{\infty} 7^{n + 1} \cdot 8^{- n} 
Sum(7^(n + 1)*8^(-n), (n, 1, oo))
The radius of convergence of the power series
Given number:
7n+1⋅8−n7^{n + 1} \cdot 8^{- 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=7n+1a_{n} = 7^{n + 1}
and
x0=−8x_{0} = -8
,
d=−1d = -1
,
c=0c = 0
then
1R=∞~(−8+lim⁡n→∞(7−n−2⋅7n+1))\frac{1}{R} = \tilde{\infty} \left(-8 + \lim_{n \to \infty}\left(7^{- n - 2} \cdot 7^{n + 1}\right)\right)
Let's take the limit
we find
False

False

R=0R = 0
The rate of convergence of the power series
1.07.01.52.02.53.03.54.04.55.05.56.06.5050
The answer [src] 
49
4949 
49
Numerical answer [src] 
49.0000000000000000000000000000
49.0000000000000000000000000000
The graph
Sum of series (7^(n+1))(8^-n)

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