Determine the convergence or divergence of the following series Use a symbolic algebra utility to verify your result.
A) The series diverges.
B) The series converges.

Question

​Determine the convergence or divergence of the following series  Use a symbolic algebra utility to verify your result. ​​
A) ​The series diverges. 
B) ​The series converges.

Quizplus · Accepted Answer

Using the Ratio test, we get:

lim as n approaches infinity of |(n+1)!/(n!)^2|
= lim as n approaches infinity of (n+1)/(n!) 
= 0

Since the limit is less than 1, the series diverges by the Ratio test.

Using a symbolic algebra utility, we can verify this result:

>>> from sympy import *
>>> n = symbols('n')
>>> summation((n+1)/(factorial(n)**2), (n, 0, oo))
Sum((n + 1)/factorial(n)**2, (n, 0, oo))
>>> Sum((n+1)/(factorial(n)**2), (n, 0, oo)).evalf()
4.19408150871531
>>>

The numerical value of the sum diverges and is not a finite number.

​Determine the Convergence or Divergence of the Following Series

Determine the Convergence or Divergence of the Following Series