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limit of real number sequence
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(Definition)
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An endless real number sequence
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(1) |
has the real number $L$ as its limit, if the distance between $L$ and $a_n$ can be made smaller than an arbitrarily small positive number $\varepsilon$ by chosing the ordinal number $n$ of $a_n$ sufficiently great, i.e. greater than a number $N$ (the size of which depends on the value of $\varepsilon$ ); accordingly $$|L-a_n| < \varepsilon \quad \mbox{when} \quad n > N.$$ Then we may denote
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(2) |
or equivalently
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(3) |
Remark 1. One should not think, that $a_n = L$ when $n = \infty$ . The symbol ``$\infty$ '' 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 $$\lim_{n\to\infty}\frac{2n}{n\!+\!1} \;=\; 2.$$ It's a question of that the real number sequence $$\frac{2}{2},\;\frac{4}{3},\;\frac{6}{4},\;\ldots$$ has the limit value 2 (e.g. the nine hundred ninety-ninth member $\frac{1998}{1000} = 1.998$ is already ``almost'' 2!). For justificating the result, let $\varepsilon$ be an arbitrary positive number, as small as you want. Then
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(4) |
when $n$ is chosen so big that
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(5) |
The condition (5) is obtained from (4) by solving this inequality for $n$ . In this case, we have $N = \frac{2}{\varepsilon}\!-\!1$ .
Example 2. The so-called decimal expansions, i.e. endless decimal numbers, such as
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(6) |
are, as a matter of fact, limits of certain real number sequences. E.g. the last of these is related to the sequence
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(7) |
which may be also written as $$1\!-\!\frac{1}{10},\; 1\!-\!\frac{1}{10^2},\; 1\!-\!\frac{1}{10^3},\;\ldots $$ The limit of (7) is 1. Actually, if $\varepsilon > 0$ , the distance between 1 and the $n^\mathrm{th}$ member of (7) is $$\left|1-\left(1\!-\!\frac{1}{10^n}\right)\right| = \frac{1}{10^n} < \varepsilon,$$ when $10^n > \frac{1}{\varepsilon}$ , i.e. when $n > -\log_{10}\varepsilon = N$ .
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 $$\lim_{n\to\infty}\frac{1\!+\!(-1)^n}{2n} = 0$$ and every other member of the sequence in question is 0.
There are sequences that have no limit at all, for example $1,\,-1,\,1,\,-1,\,1,\,-1,\,\ldots$ . Some real number sequences (1) have the property, that the member $a_n$ may exeed every beforehand given real number $M$ if one takes $n$ greater than some value $N$ (which depends on $M$ ): $$a_n > M \quad \mbox{when} \quad n > N.$$ Then we write $$\lim_{n\to\infty}a_n \;=\; \infty.$$ Similarly, the sequence (1) may be such that for each positive $M$ there is $N$ such that $$a_n < -M \quad \mbox{when} \quad n > N,$$ and then we write $$\lim_{n\to\infty}a_n \;=\;
-\infty.$$ E.g. $$\lim_{n\to\infty}n^2 \;=\; \infty, \quad \lim_{n\to\infty}(1\!-\!n) \;=\; -\infty.$$
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"limit of real number sequence" is owned by pahio.
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Cross-references: property, finite, decimal numbers, decimal expansions, inequality, hundred, necessary, number, positive, distance, sequence, real number
There are 67 references to this entry.
This is version 13 of limit of real number sequence, born on 2008-11-19, modified 2009-02-14.
Object id is 11263, canonical name is LimitOfRealNumberSequence.
Accessed 2042 times total.
Classification:
| AMS MSC: | 40A05 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences) |
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Pending Errata and Addenda
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