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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 <a_{n}> is said to converge to a real number L if a_{H} 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 = lim_{n→∞} a_{n}. Figure 9.1.2 A sequence which does not converge to any real number is said to diverge. If a_{H} is positive infinite for all positive infinite hyperintegers H, the sequence is said to diverge to ∞, and we write lim_{n→∞} a_{n} = ∞. 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 a_{H} for infinite H. The definition gives us some convenient working rules. (1) If a_{H} is infinitely close to L for all H, the sequence converges to L. (2) If we can find a_{H} and a_{K} which are not infinitely close to each other, the sequence diverges. (3) If at least one a_{H} is infinite, the sequence diverges. (4) If all the a_{H} are positive infinite, the sequence diverges to ∞.
PROOF
Let H be positive infinite. For some K,
Then a_{H} = K / 10^{H} a_{H} ≤ π ≤ a_{H} + 1/ 10^{H} But 1/10^{H} is infinitesimal, so a_{H} ≈ π.
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 xcoordinate is a positive integer (Figure 9.1.3). If lim_{n→∞} f(x) = L, then lim_{n→∞} f(n) = L because f(H) x L for any positive infinite H. Figure 9.1.3


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