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(-8)^(n+1)/(n+1)!

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



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The solution

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  oo           
____           
\   `          
 \        n + 1
  \   (-8)     
  /   ---------
 /     (n + 1)!
/___,          
n = 1          
n=1(8)n+1(n+1)!\sum_{n=1}^{\infty} \frac{\left(-8\right)^{n + 1}}{\left(n + 1\right)!}
Sum((-8)^(n + 1)/factorial(n + 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
(8)n+1(n+1)!\frac{\left(-8\right)^{n + 1}}{\left(n + 1\right)!}
It is a series of species
an(cxx0)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+limnanan+1cR^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}
In this case
an=(8)n+1(n+1)!a_{n} = \frac{\left(-8\right)^{n + 1}}{\left(n + 1\right)!}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=limn(8n28n+1(n+2)!(n+1)!)1 = \lim_{n \to \infty}\left(8^{- n - 2} \cdot 8^{n + 1} \left|{\frac{\left(n + 2\right)!}{\left(n + 1\right)!}}\right|\right)
Let's take the limit
we find
False

False
The rate of convergence of the power series
1.07.01.52.02.53.03.54.04.55.05.56.06.5-500500
The answer [src]
     -8
7 + e  
e8+7e^{-8} + 7
7 + exp(-8)
Numerical answer [src]
7.00033546262790251183882138913
7.00033546262790251183882138913
The graph
Sum of series (-8)^(n+1)/(n+1)!

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