Here is a solution using an index-based loop to create slices.

for i in range(len(word)):
    print word[ :1+i]

Here is a solution that builds up the pieces character-by-character.

latest = ''	
for c in word:
    latest += c      # add newest letter to end
    print latest

fact = 1
for val in range(2,k+1):   # up to but not including k+1
    fact = fact * val

e = 0.0
for k in range(n):
    # compute factorial for k as above
    fact = 1
    for val in range(2,k+1):   # up to but not including k+1
        fact = fact * val
    # now add in a new term
    e = e + 1.0/fact           # note use of float 1.0 in numerator

We can do even better by not recomputing the factorial from scratch each time, but instead basing it on the previous iteration, as follows.

e = 1.0         # let's add in first term by hand for 1/0!
factorial = 1
for k in range(1,n):   # other n-1 terms
    # update factorial
    factorial = factorial * k
    # add new term
    e = e + 1.0/factorial

original = raw_input('Enter numeric string: ')
base = int(raw_input('What base is the number? '))
place = 1                    # we start with the one's digit
total = 0
original = original[ : :-1]  # reverse for convenience
for digit in original:
    total = total + place * int(digit)
    place = place * base
print total

byLast = []
for person in people:
    pieces = person.split()
    byLast.append( (pieces[1], pieces[0]) )    # (lastname,firstname)
byLast.sort()
for person in byLast:
    print person[1] + ' ' + person[0]          # back to 'firstname lastname'