Calculus: Analytic Geometry and Calculus, with Vectors

140 Functions, limits, derivatives

15 Try to make friends of the contents of the preceding problems by proving
that

(1)

()+(

kn 1)-\nk

when n, k - 1, and n - k are not negative integers. Remark: As some people

learn while studying algebra, the ordinary binomial coefficients (in which n and

k are integers for which 0 < k < n) are the coefficients appearing in the formulas

(2) (a + b)° = 1

(3) (a + b)1 = a + b

(4) (a+b)2=a2+2ab+b2

(5) (a + b)3 = a3 + 3a22b + 3ab2 + b3

and, in general, in the binomial formula

(6) (a + b)n =(Olanbo + (1n) an-ib +

C2/

an-2b2 + .. +(nla°b".

With the aid of (1), it is easy to fill in the rows of the Pascal triangle

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

which displays binomial coefficients. The sum of two consecutive elements of

one row gives the element that lies below the space between them, and more

rows of the Pascal triangle are easily written.

16 We can feel sure that if x > 1, then xn must be large whenever n is large,

but it is nevertheless worthwhile to be able to prove the precise version of the

statement. When x > 1, there is a positive number h such that

(1) x=1+h;

in fact, h = x - 1. Observe that

(2) x2=1+2h+h2>1+2h

(3) x3= 1+3h+3h2+h3>1+3h

and that the binomial formula shows that

(4) xn>1+nh

when n >= 2. It follows that if M is a given number and we choose a number N

such that N > 2 and N > M/h, then we will have

(5) x">1+nh>nh>M

whenever n > N. Therefore,

(6) lim x" _ 00 (x > 1).

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