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(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           
$$\sum_{n=1}^{\infty} 7^{n + 1} \cdot 8^{- n}$$
Sum(7^(n + 1)*8^(-n), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$7^{n + 1} \cdot 8^{- n}$$
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
$$a_{n} = 7^{n + 1}$$
and
$$x_{0} = -8$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-8 + \lim_{n \to \infty}\left(7^{- n - 2} \cdot 7^{n + 1}\right)\right)$$
Let's take the limit
we find
False

False

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

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