Mister Exam

Other calculators

3^(n+1)/factorial(n+1)
Sum of series/ 3^(n+1)/factorial(n+1)

Sum of series 3^(n+1)/factorial(n+1)


∑
=

The solution

You have entered [src] 
  oo          
____          
\   `         
 \      n + 1 
  \    3      
  /   --------
 /    (n + 1)!
/___,         
n = 1
∑n=1∞3n+1(n+1)!\sum_{n=1}^{\infty} \frac{3^{n + 1}}{\left(n + 1\right)!} 
Sum(3^(n + 1)/factorial(n + 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
3n+1(n+1)!\frac{3^{n + 1}}{\left(n + 1\right)!}
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+1(n+1)!a_{n} = \frac{3^{n + 1}}{\left(n + 1\right)!}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=lim⁡n→∞(3−n−2⋅3n+1∣(n+2)!(n+1)!∣)1 = \lim_{n \to \infty}\left(3^{- n - 2} \cdot 3^{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
1.07.01.52.02.53.03.54.04.55.05.56.06.5020
The answer [src] 
      3
-4 + e
−4+e3-4 + e^{3} 
-4 + exp(3)
Numerical answer [src] 
16.0855369231876677409285296546
16.0855369231876677409285296546
The graph
Sum of series 3^(n+1)/factorial(n+1)

    Examples of finding the sum of a series

    © Mister Exam – Calculator