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Homelimit of real number sequence

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An endless real number sequence

$\displaystyle a_{1},\,a_{2},\,a_{3},\,\ldots$ | (1) |

has the real number $A$ as its limit, if the distance between $A$ 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

$|A-a_{n}|<\varepsilon\quad\mbox{when}\quad n>N.$ |

Then we may denote

$\displaystyle\lim_{{n\to\infty}}a_{n}\;=\;A$ | (2) |

or equivalently

$\displaystyle a_{n}\to A\quad\mbox{as}\quad n\to\infty.$ | (3) |

Remark 1. One should not think, that $a_{n}=A$ 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

$\displaystyle\left|2-\frac{2n}{n\!+\!1}\right|=\left|\frac{2n\!+\!2}{n\!+\!1}-% \frac{2n}{n\!+\!1}\right|=\left|\frac{2}{n\!+\!1}\right|=\frac{2}{n\!+\!1}<\varepsilon,$ | (4) |

when $n$ is chosen so big that

$\displaystyle n>\frac{2}{\varepsilon}\!-\!1.$ | (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

$\displaystyle 3.14159265\ldots\,=\,\pi,\quad 0.636363\ldots,\quad 0.99999\ldots,$ | (6) |

are, as a matter of fact, limits of certain real number sequences. E.g. the last of these is related to the sequence

$\displaystyle 0.9,\;0.99,\;0.999,\;\ldots$ | (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.

# Infinite limits of real number sequences

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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## Comments

## Limits of positive real number sequences

pahio,

The limit of 2n/(n+1) as positive n goes towards infinity is 2.

0/1, 2/2, 4/3, 6/4, ... 1998/1000.

[The number 999 was a very astute choice for n.]

1.98, 1.998, 1.9998, 1.99998, gets closer and closer to 2.

Now 1.98 + .02 = 2; 1.998 + .002 = 2; 1.9998 + .0002 = 2; ... 1.9999...98 + .0000...2 = 2.

If everybody is willing I'd like to call .0000...2 the error, with the symbol lower case epsilon.

For negative n, the limit of 2n/(n-1) as negative n goes to negative infinity is -2.

Re: The sequence 1, -1, 1, -1, ... +/- 1.

The series 1 + (-1) + 1 + (-1) + 1 + (-1)... +/-1 = 1, 0, 1, 0, 1, ...; has wavy behavior, and may be hidden in other sequences.

We could also write 0.9, 0.99, 0.999, ... 0.9999...9 as (1-(10^{-n})).

Unfortunately, marie makes lots of mistakes, and she can't get through a page of calculations without messing up the signs. Worse yet, she'd think of a possibility like (1-(10^{-(-n)}).

Thank you for all the fruitful ideas.

- marie

## Re: Limits of positive real number sequences

Dear Marie,

A little sign error: We have

2n/(n+1) = 2/(1+1/n) --> 2/(1+0) = 2 as n --> -OO.

Jussi

## Re: Limits of positive real number sequences

agree with you