given \sum_(n=0)^(\infty ) ((2n)!)/((5000)^(n)) write a python code to the following questions:

Question

a) print the first 16 terms of the series (from a0 to a15). Based on your output, do you expect the series to converge or diverge?

b) Apply the ration test to the series, i.e., compute |an+1/an| and \lim_(n->\infty ) |an+1/an|

c) what does your answer to part b tell you about the series?

Answer 1

a.)

for n in range(16):

print(factorial(2*n) / 5000**n)

Based on the results you see, it will look like it converges

b.)

|a_n+1/a_n| = | ((2n+2)!/5000^(n+1)) / ((2n)!/5000^n) | = | (2n+2)!/(2n)! * 5000^n/5000^(n+1) | = | (2n+1)(2n+2)/5000 |

In the case as n goes to ∞, this ratio goes to 4∞²/5000 = ∞.

c.) The series actually diverges