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Limit of a sequence

the sequence limit

Let’s first try to notice some important characteristics of sequences. For sequence:

1, 2, 3, 4,... , n, ...

we can see that members of it are constantly growing, and for sequence

 

1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4},...,\frac{1}{n},...

 

we can see that the members are constantly reducing.

Of course, there are many sequences where we can’t find any of that regularities such as alternating sequences.


Sequence of real numbers a_1, a_2,..., a_n,...  will monotonic increase if a_n \ge a_{n - 1} for every n \in \mathbb{N}.

Sequence of real numbers a_1, a_2,..., a_n,...  will  monotonic decrease if a_n \le a_{n - 1} for every n \in \mathbb{N}.

The sequence is strictly monotonic if sign \le and \ge is sign > or <.


Notice that for a proof that a sequence is monotonic increasing it is enough to show that a_{n + 1} - a_n \ge 0

If a_{n + 1} - a_n, then the sequence is decreasing, and if a_{n + 1} - a_n changes its sign for some values of n, then the sequence is not monotonic sequence.

monotonic increasing

monotonic decreasing

Example 1. Is sequence whose general member is a_n = \frac{n - 1}{n + 1} monotonic?

Let’s find the difference a_{n + 1} - a_n  = \frac{n}{n + 2} - \frac{n - 1}{n + 1} = \frac{2}{(n + 2)(n + 1)}

Since n \in \mathbb{N}, \frac{2}{(n + 2)(n + 1)} > 0 or a_{n + 1} > a_n which means that the sequence is monotonic increasing.

Let’s notice some more properties. For example, what do you notice for a sequence given with general member a_n = 1 + \frac{1}{n}

Probably, just from a look of it, nothing much. Let’s write some of its first members.

1, \frac{3}{2}, \frac{4}{3}, \frac{5}{4}, \frac{6}{5}, \frac{7}{6},...

We can notice that the members of this sequence never go over 2, but never decrease below 1. This means that for every member of this sequence is valid that:

1 < a_n \le 2

Here we came to second important sequence characteristic:


Sequence a_1, a_2,..., a_n,... is bounded if there exist numbers m and M such that for every a_n it is true that:

m \le a_n \le M.

Number m is the lower bound and M is the upper bound of the set S = \{ a_1, a_2,..., a_n,...\}.

We say that the sequence is bounded if we can find its lower and upper bound. Every member of bounded sequence are found in segment [m, M].


If we know that a sequence is bounded, we can find infinitely many upper and lower bounds.

 

Example 2. For the following sequences determin  bound?

a) a_n = \frac{1}{n}

Let’s write again first few members.

1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6},...

monotonic decrease example

We can clearly see that one of the lower bounds is zero and one of the upper is 1.

0 < a_n \le 1

b) a_n = (- \frac{1}{2})^n

Don’t let this minus frighten you. Just write first few members and conclude what will happen to this sequence.

1, -\frac{1}{2}, \frac{1}{4}, \frac{1}{16}, -\frac{1}{32}, \frac{1}{64},...

From here we can see that this sequence is bounded by the first two members. If we’d continue writing these members they would be approaching zero.

This means that -\frac{1}{2} < (-\frac{1}{2})^n < \frac{1}{4}.

c) a_n = (- 1)^n n

Again, we’ll write down first few members.

-1, 2, -3, 4, -5, 6, -7...

writting-down-monotonic-increase-and-decrease-from-example

This sequence has no bounds.

One of the most important questions we can ask when observing infinite sequence (a_n) is:

What happens to the members of a sequence for great values of n?

 

Let’s observe sequence that is given with his general member a_n = 1 - \frac{1}{10^n}.

a_1 = 0.9 a_2 = 0.99 a_3 = 0.99

We can see that the larger the n is, the value of sequences member is closer to 1.

We say that number 1 is the limit value or limit of a sequence a_n. This statement is noted as:

lim_{n -> \infty} (1 - \frac{1}{10^n}) = 1

To be accurate this is read:

The limit value of sequence (1 - \frac{1}{10^n}) = 1 when n tends to infinity is equal to 1.

What about alternating sequences? They also can have limits. Let’s observe sequence a_n = (- 1)^n \frac{1}{n}

a_1 = - 1a_2 = \frac{1}{2},  a_3 = - \frac{1}{3},  a_4 = \frac{1}{4},  a_5 = - \frac{1}{5}, a_6 = \frac{1}{6}, …

This sequence is bounded, - 1 \le a_n \le 1 for every n. When n grows members of this sequences are approaching zero and the distance of nth member from zero gets smaller and smaller.

number line sequences greater values

This means that

lim_{n -> \infty} (- 1)^n * \frac{1}{n} = 1


For a sequence a_n of real numbers is said to be convergent if there exists real number a such that sequence a_n tends to this number when n grows infinitely.

We say that a is the limit of a sequence and write lim_{n -> \infty} (a_n) = a.


Can we define limit value in some other way?

If we know that that a limit is a number to which the sequence is tending we can rephrase that sentence in some other way. If <em>a is a limit of a sequence, then as n grows, the distance between the member of that sequence and number <em>a is decreasing.

This means that we can find as small number \varepsilon as we’d like, such that the distance from the members of the sequence to number <em>a is smaller than ε for almost every (all but finitely many) member of the sequence or:

| a_n - a | < \varepsilon

Geometrically watched, condition | a_n - a | < \varepsilon means that there is as small interval as we’d like < a - \varepsilon, a + \varepsilon > that contains every member but finitely many.


Real number <em>a is the limit of a sequence of real numbers a_n if for every \varepsilon > 0 exists natural number n_0 such that for every n > n_0 it is valid that:

| a_n - a | < \varepsilon


For a sequence that is not convergent we say that it is divergent.

Let’s check that the sequence a_n, a_n = \frac{n - 1}{n} has limit that is equal to 1 using this second definition. We can imagine a small number \varepsilon, for example \varepsilon = 0.001.

\mid a_n \mid  \le \varepsilon \rightarrow \mid \frac{n - 1}{n} - 1 \mid \le 0.001 \rightarrow \rightarrow = 1000

From here, we can conclude that for every n \ge 1000 the distance of number 1 to the member of the sequence will be lesser than 0.001; every member of a sequence (except for the first 1000) is in environment of number 1.

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