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(3^n-1)/n!

  • How to use it?

  • Sum of series:
  • 1/(n(n+2)) 1/(n(n+2))
  • (3^n-1)/n! (3^n-1)/n!
  • (2n-1)/2^n (2n-1)/2^n
  • 1/(n+1)! 1/(n+1)!

  • Identical expressions

  • (three ^n- one)/n!
  • (3 to the power of n minus 1) divide by n!
  • (three to the power of n minus one) divide by n!
  • (3n-1)/n!
  • 3n-1/n!
  • 3^n-1/n!
  • (3^n-1) divide by n!

  • Similar expressions

  • (3^n+1)/n!
Sum of series/ (3^n-1)/n!

Sum of series (3^n-1)/n!


∑
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The solution

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

False
The rate of convergence of the power series
0.06.00.51.01.52.02.53.03.54.04.55.05.5020
The answer [src] 
      3
-E + e
−e+e3- e + e^{3} 
-E + exp(3)
Numerical answer [src] 
17.3672550947286225055682421832
17.3672550947286225055682421832
The graph
Sum of series (3^n-1)/n!

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