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limit of real number sequence∗ pahio† 2013-03-22 2:02:13 An endless real number sequence a1 , a2 , a3 , . . . (1) has the real number L as its limit, if the distance between L and an can be made smaller than an arbitrarily small positive number ε by chosing the ordinal number n of an sufficiently great, i.e. greater than a number N (the size of which depends on the value of ε); accordingly |L − an | < ε when n > N. Then we may denote lim an = L n→∞ (2) or equivalently an → L as n → ∞. (3) Remark 1. One should not think, that an = L when n = ∞. The symbol “∞” represents no number, one cannot set it for the value of n. It’s only a question of allowing n to exceed any necessary value. Example 1. Using the notation (2) we can write a result lim n→∞ 2n = 2. n+1 It’s a question of that the real number sequence 2 4 6 , , , ... 2 3 4 ∗ hLimitOfRealNumberSequencei created: h2013-03-2i by: hpahioi version: h41263i Privacy setting: h1i hDefinitioni h40A05i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1 1998 has the limit value 2 (e.g. the nine hundred ninety-ninth member 1000 = 1.998 is already “almost” 2!). For justificating the result, let ε be an arbitrary positive number, as small as you want. Then 2 − 2n = 2n+2 − 2n = 2 = 2 < ε, (4) n+1 n+1 n+1 n+1 n+1 when n is chosen so big that n> 2 −1. ε (5) The condition (5) is obtained from (4) by solving this inequality for n. In this case, we have N = 2ε −1. Example 2. The so-called decimal expansions, i.e. endless decimal numbers, such as 3.14159265 . . . = π, 0.636363 . . . , 0.99999 . . . , (6) are, as a matter of fact, limits of certain real number sequences. E.g. the last of these is related to the sequence 0.9, 0.99, 0.999, . . . (7) which may be also written as 1− 1 1 1 , 1− 2 , 1− 3 , . . . 10 10 10 The limit of (7) is 1. Actually, if ε > 0, the distance between 1 and the nth member of (7) is 1 − 1− 1 = 1 < ε, n 10 10n when 10n > 1ε , i.e. when n > − log10 ε = N . The endless decimal notations (6) and others are, in fact, limit notations — no finite amount of decimals in them suffices to give their exact values. Remark 2. In both of the above examples, no of the sequence members was equal to the limit, but it does not need always to be so; thus for example 1+(−1)n =0 n→∞ 2n lim and every other member of the sequence in question is 0. 2 Infinite limits of real number sequences There are sequences that have no limit at all, for example 1, −1, 1, −1, 1, −1, . . .. Some real number sequences (1) have the property, that the member an may exeed every beforehand given real number M if one takes n greater than some value N (which depends on M ): an > M when n > N. Then we write lim an = ∞. n→∞ Similarly, the sequence (1) may be such that for each positive M there is N such that an < −M when n > N, and then we write lim an = −∞. n→∞ E.g. lim n2 = ∞, n→∞ lim (1−n) = −∞. n→∞ 3