Calculus: Analytic Geometry and Calculus, with Vectors

3.3 Unilateral limits and asymptotes 139

which is applicable when n is a positive integer, show that 1! = 1, 2! = 2,
3! = 6, 4! = 24, 5! = 120, 6! = 720, and 7! = 5040. Then give a full statement
of the reason or reasons why it is true that, when z is a positive integer,

(2) z! = lim 1.2.3 z

() 3 z! = lim 1.2.3... z(z + 1)(z + 2) ... (z + n)

n-» (z + 1) (z + 2) ... (z + n)

4

__

z

n!n- n+In-I-2 n + z

( ) ' n .(z+1)(z+2). .. (z+n) n n n

(^5) Z! = lim
n!-n:
()

,,.. (z + 1)(z + 2) ... (z +

n)

Remark: To show that the above manipulations serve a useful purpose, we take

a little mental excursion. A complex number z is a number of the form x + iy,

where x and y are real numbers and i is the imaginary unit for which i2 = -1.

While this book neither develops nor uses the algebra and calculus of complex

numbers, we remark that x + iy is the real number x if y = 0 and that x + iy

is a real integer if y = 0 and x is a real integer. We are now ready to look at

(5). We have seen that (5) is correct if z is a positive integer and the definition

(1) is applicable. While proof of the fact lies far beyond our present capabilities,

it can be proved that the limit in the right member of (5) exists and is a complex

number whenever z is a complex number which is not a negative integer. More-

over, when z is a complex number which is not a negative integer, z! is defined

to be this limit. It follows from the definition that z! is a real number whenever

z is a real number which is not a negative integer. Carl Friedrich Gauss (1777-

1855), who had the habit of knowing how things should be done, made very

effective use of (5). The index can always show us where this and other informa-

tion about factorials is concealed.

12 If the preceding problem and remark have been digested, prove that

0! = 1. Remark: Proof of the more esoteric facts that (-i)! = -\/;and (i.)! _

-/2 will not be too difficult when more mathematics of the right kind has been

learned.

13 Observe that 8! = 8(7!). Then, assuming that the limits exist, prove that

1 n!n:+r

.(z+1+1)(z+1+2) ... (z+1+n)

_ (z + 1) lim n!n°

n-W (z + 1)(z + 2)... (z + n)

Finally, use the remark of Problem 11 to prove that

(z + 1)! _ (z + 1)(z!)

when z is not a negative integer.

14 For what pairs of numbers n and k does it make sense to define the binomial

coefficient function by the formula

nll_ n!?

Ck/ k!(n - k)!'

Hint: If necessary, read Problem 11. Ins.: When n, k, and n - k are numbers

(real or complex) that are not negative integers.