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(6*n-7)/2^n
Sum of series/ (6*n-7)/2^n

Sum of series (6*n-7)/2^n


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The solution

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  oo         
____         
\   `        
 \    6*n - 7
  \   -------
  /       n  
 /       2   
/___,        
n = 1
∑n=1∞6n−72n\sum_{n=1}^{\infty} \frac{6 n - 7}{2^{n}} 
Sum((6*n - 7)/2^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
6n−72n\frac{6 n - 7}{2^{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=6n−7a_{n} = 6 n - 7
and
x0=−2x_{0} = -2
,
d=−1d = -1
,
c=0c = 0
then
1R=∞~(−2+lim⁡n→∞∣6n−76n−1∣)\frac{1}{R} = \tilde{\infty} \left(-2 + \lim_{n \to \infty} \left|{\frac{6 n - 7}{6 n - 1}}\right|\right)
Let's take the limit
we find
False

R=0R = 0
The rate of convergence of the power series
1.07.01.52.02.53.03.54.04.55.05.56.06.55-5
The answer [src] 
5
55 
5
Numerical answer [src] 
5.00000000000000000000000000000
5.00000000000000000000000000000
The graph
Sum of series (6*n-7)/2^n

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