#note: should NOT import any packages or variables for this problem

def double_fact(n): #this uses a while loop

''' return the double factorial of an integer n usually denoted as n!!, (where n>=-1),

which is defined as

(-1)!! = 1 and 0!! = 1,

for n>0: n!! = n*(n-2)*(n-4)* ...*k, where k=1 if n is odd, and k=2 if n is even.

Requirements;

1. you are not allowed to call any external functions;

2. you need to use a 'while' loop, do not write a recursive function

'''

def double_fact2(n):

""" return and requirement are the same as above, except that you need to use a 'for' loop here"""

def Ramanujan(tol=1e-14):

'''compute and return the converged sum of

1/1!! + 1/3!! + 1/5!! + 1/7!! + ... + 1/(2n-1)!! + ... ,

you can consider the sum as convergent when the last term you add to the sum becomes

Requirements:

1. you should call the double_fact() or double_fact2() function you write above,

2. use 'for' loop with 10**9 as the loop upper bound, break out of loop when the sum converges.

'''

def Ramanujan2(tol=1e-14):

'''compute and return the converged sum of

1/1!! + 1/3!! + 1/5!! + 1/7!! + ... + 1/(2n-1)!! + ... ,

you can consider the sum as convergent when the last term you add to the sum becomes

Requirements:

1. you should NOT call the double_fact() or any other factorial function, but

instead uyou should se update: i.e., you should compute (n+2)!! as n!!*(n+2),

reusing the n!! that has already been computed at a previous iteration.

2. for this problem you can use either 'for' or 'while' loop

'''

## do not modify the test code below this line

if __name__=='__main__':

#test on n = -1, 0, 1, 2, 3, ..., 11

DF0 = [ double_fact(n) for n in range(-1,12) ]

DF0_ext =[1, 1, 1, 2, 3, 8, 15, 48, 105, 384, 945, 3840, 10395]

if DF0 != DF0_ext:

print(' ***** double_fact() failed the test ***** ')

print(' you got DF0 =', DF0)

print(' should be ', DF0_ext)

else:

print(' Great, double_fact() passed the test ')

DF02 = [ double_fact2(n) for n in range(-1,12) ]

if DF02 != DF0_ext:

print(' ***** double_fact2() failed the test ***** ')

print(' you got DF02 =', DF02)

print(' should be ', DF0_ext)

else:

print(' Great, double_fact2() passed the test ')

#test on other n

DF1 = [ double_fact(n) for n in range(10, 18) ]

DF1_ext = [3840, 10395, 46080, 135135, 645120, 2027025, 10321920, 34459425]

if DF1 != DF1_ext:

print(' ***** double_fact() failed the test ***** ')

print(' you got DF1 =', DF1)

print(' should be ', DF1_ext)

else:

print(' Great, double_fact() passed another test ')

DF2 = [ double_fact2(n) for n in range(20, 26) ]

DF2_ext = [3715891200, 13749310575, 81749606400, 316234143225, 1961990553600, 7905853580625]

if DF2 != DF2_ext:

print(' ***** double_fact2() failed the test ***** ')

print(' you got DF2 =', DF2)

print(' should be ', DF2_ext)

else:

print(' Great, double_fact2() passed another test ')

## test on the sum

sumR=[1.4106861040194372, 1.4106861345664234, 1.4106861346391544, 1.4106861346423165, 1.4106861346424477]

for i, tol in enumerate([1e-6, 1e-8, 1e-10, 1e-11, 1e-14]):

sum = Ramanujan(tol=tol); sum2 = Ramanujan2(tol=tol)

if abs(sumR[i] - sum) > 2*tol:

print('***** Ramanujan ({}) test failed, your sum={}, err={} *****'.

format(tol, sum, abs(sumR[i] - sum)))

else:

print(' Great. Ramanujan ({}) test passed!'.format(tol))

if abs(sumR[i] - sum2) > 2*tol:

print('***** Ramanujan2({}) test failed, your sum={}, err={} *****'.

format(tol, sum, abs(sumR[i] - sum)))

else:

print(' Great. Ramanujan2({}) test passed!'.format(tol))

0.6 0.4 0.2 0.0 0.4 0.2 -2 -1 0 l2-2 -1 l 2 Figure 1: Plot of logistic functions P2. (40 points) Finish the python script double fact.py, which contains several subproblems: (a) Write two functions to compute the double factorials (n!!) of an input integer n, where double factorial is the same as factorial, except that its step-length is 2. For odd n, n!! = n * (n-2) * . . . * 1; F -2) * * 2; Also, 0!! = 1, and or even n, n!! n(n (-1)! 1, you can disregard other negative n's. (b) Compute the following sum in two different ways (2i- 1 Follow the instructions in double-fact.py for requirements of the d