## Definition: Limit of a Sequence

<n!> is an important sequence. Its first few terms are

1, 2, 6, 24, 120, 720,....

By convention, 0! is defined by 0! = 1.

DEFINITION

An infinite sequence <an> is said to converge to a real number L if aH is infinitely close to L for all positive infinite hyperintegers H (Figure 9.1.2). L is called the limit of the sequence and is written

L = limn→∞ an.

Figure 9.1.2

A sequence which does not converge to any real number is said to diverge. If aH is positive infinite for all positive infinite hyperintegers H, the sequence is said to diverge to ∞, and we write

limn→∞ an = ∞.

Sequences can diverge to -∞, and also diverge without diverging to ∞ or to -∞.

Throughout this chapter, H and K will always be used for positive infinite hyperintegers. One can often determine whether or not a sequence converges by examining the values of aH for infinite H. The definition gives us some convenient working rules.

(1)    If aH is infinitely close to L for all H, the sequence converges to L.

(2)    If we can find aH and aK which are not infinitely close to each other, the sequence diverges.

(3)    If at least one aH is infinite, the sequence diverges.

(4)    If all the aH are positive infinite, the sequence diverges to ∞.

 Example 1 (Continued)
 Example 2 (Continued)
PROOF

Let H be positive infinite. For some K,

Then

aH = K / 10H aH ≤ π ≤ aH + 1/ 10H

But 1/10H is infinitesimal, so aH ≈ π.

 Example 3 (Continued): n Factorial

PROOF

For any n > 1, we have

(n - 1)! ≥ 1, n! = n - (n - 1)! ≥ n.

Therefore for positive infinite H, H! ≥ H is positive infinite.

Given a function f(x) defined for all x ≥ 1, we can form the sequence f(1), f(2),..., f(n),....

The graph of the sequence < f(n)> is the collection of dots on the curve y = f(x) where the x-coordinate is a positive integer (Figure 9.1.3).

If limn→∞ f(x) = L, then

limn→∞ f(n) = L

because f(H) x L for any positive infinite H.

Figure 9.1.3

 Example 4
 Example 5
 Example 6

Last Update: 2006-11-07