Unizor - Derivatives - Limit of Sequence - Definition and Properties

Notes to a video lecture on http://www.unizor.com


Sequence Limit -
Definition and Properties

Please refer to lectures on sequence limits in the "Limits" chapter of Algebra subject of this course.
Here is a brief reminder of a definition and basic properties of sequence limits.

A sequence S={a_n} is an infinite countable ordered set of real numbers, where for each natural number n exists one and only one element of this set a_n.

Real number L is a limit of a sequence {a_n}, if for any (however small) ε > 0 there exists order number N such that
|L - a_n| ≤ ε for any n ≥ N.

The requirement of existence of an order number N with corresponding sequence term being closer to a limit than any chosen distance ε, however small we choose it, assures that elements of a sequence eventually become, as we say, infinitely close to a limit.
The requirement of absolute value of a distance between limit L and elements of a sequence a_n to be not greater than ε for any n ≥ N assures that, once a sequence get sufficiently close to a limit, it will stay not farther from it.

A sequence that has a limit is called convergent, it converges to its limit.

Let's address some simple properties of limits. All of them were proven in the lectures about sequence limit in the Algebra subject of this course. We strongly recommend to review these proofs in that lecture.

Theorem 1
A convergent sequence is bounded, that is there are two numbers, lower and upper bounds, such that all elements of this sequence are not less than lower and not greater than upper bound.
Symbolically,
{a_n}→L
⇒ ∃ A, B ∀ n: A ≤ a_n ≤ B

Theorem 2
A convergent sequence, multiplied by a factor, converges to a limit that is equal to a limit of an original sequence, multiplied by this factor.
Symbolically,
{a_n}→L
⇒ {K·a_n}→K·L

Theorem 3
A sum of two convergent sequences converges to a limit that is equal to a sum of the limits of these two sequences.
Symbolically,
{a_n}→L; {b_n}→M
⇒ {a_n+b_n}→L+M

Theorem 4
A product of two convergent sequences converges to a limit that is equal to a product of limits of these two sequences.
Symbolically,
{a_n}→L; {b_n}→M
⇒ {a_n·b_n}→L·M

Theorem 5
An inverse of a convergent sequence, that has a non-zero limit, converges to a limit that is equal to an inverse of its limit.
Symbolically,
{a_n}→L; L≠0
⇒ {1/a_n}→1/L

Theorem 6
A ratio of two convergent sequences converges to a limit that is equal to a ratio of limits of these two sequences, provided the limit of denominator is not zero.
Symbolically,
{a_n}→L; {b_n}→M; M≠0
⇒ {a_n/b_n}→L/M

Here are examples of simple sequences that have limits:
{1/n}→0
{(n+1)/n}→1
{(12n²+3n)/(5n²−5)}→12/5