Unizor - Derivatives - Infinity

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


Sequence Limit - Infinity

We use the term infinity rather casually, understanding that this is a large, very large, larger than anything quantity.
This lecture is about a concept of infinity as it is understood by mathematicians.

First of all, let's agree that there is no such number or such quantity as infinity in classical math. There are some advanced parts of higher levels of math, where infinity is introduced as a concrete object, but it is beyond the scope of this course. So, for our purposes infinity is not a number or quantity. What is it then?

It is a short form of specifying the directional and limitless behavior of a sequence.

Consider a sequence {X_n} that grows boundlessly. That is, for any, however large, number Aexists an order number N such that all members of this sequence with order numbers not less than N are not less than number A.
Using symbols ∀ ("for all" or "for any") and ∃ ("exist"), this can be symbolically written as follows:
∀ A ∃ N: n ≥ N ⇒ X_n ≥ A

For any sequence that behaves in this manner we may say thatit's limit is infinity.
Sometimes we add a characteristic "positive" to a word infinity, if it helps to better understand the behavior of a sequence and to differentiate it from negative infinity described below.
So, the expression about infinity(or positive infinity) being a limit of some sequence cannot be considered absolutely rigorous and it just means that the sequence is boundlessly grows in a sense described above. It would be better to use the term infinitely growing sequence than mentioning the word "limit" for such cases. Also not advisable to use the term "convergent" for these sequences, this is reserved for sequences convergent to real numbers.
Example of such infinitely growing sequence:
{X_n = 2ⁿ}

Similarly, we can introduce a sequence that boundlessly "grows" (in a sense of absolute value, while being negative) to a negative infinity.
The description of this property is analogous to a case withpositive infinity.
If for any negative number A, however large by absolute value, exists an order number Nsuch that all members of this sequence with order numbers not less than N are not greater than number A, we say that the limit of this sequence isnegative infinity.
Symbolically, it can be written as
∀ A ∃ N: n ≥ N ⇒ X_n ≤ A
(notice, we don't have to specify that A is negative since we use "for any" symbol, which includes all negative numbers as well as positive)
Example of such sequence:
{X_n = log₂(1/n)} = {−log₂(n)}

So, terms infinity, positive infinity (same as infinity) andnegative infinity are legitimate mathematical characteristics of sequences that either, being positive starting at some number, grow boundlessly or, while negative after some number, grow by absolute value boundlessly.
Using these terms implies the properties described in detail above. That's why these terms can be considered as a short description of these properties. It's easier and no less rigorous to state "a sequence grows toinfinity" instead of "for any, however large, number A exists an order number N such that all members of this sequence with order numbers not less than Nare not less than number A".
Both expressions mean the same and can be used interchangeably. The former is just a lot shorter and quicker to understand. So is an expression "an infinitely growing sequence".

Examples:
1. X_n = (n²+1)/n
Let's prove that this sequence is limitlessly increasing to infinity.
Choose any boundary numberA, however large. Expression(n²+1)/n=n+1/n is monotonically increasing with an increase of n because, as nincreases by 1, 1/n decreases by a fraction of 1. Therefore, once it grows above A, it will stay above A. So, we just have to find the first member of this sequence that is above A.
Let's find natural order numberN in our sequence for a chosen boundary A by solving the inequality
(n²+1)/n ≥ A
which is equivalent to (since nis positive)
n²−An+1 ≥ 0
Expression
n²−An+1
is a quadratic polynomial of nwith discriminant A²−4, which is positive for large A (any value greater than 2).
This quadratic polynomial limitlessly grows with its argument n.
For any large number A there are two solutions to a quadratic equation n²−An+1 = 0:
n₁ = (A−√A²−4)/2 and
n₂ = (A+√A²−4)/2
For any N greater or equal to a larger of these two solutions n₂the inequality we need will be true.
Therefore, for any number A, however large, the members of our sequence {(n²+1)/n} will be greater or equal to this A as long as the order number n is greater than or equal to
(A+√A²−4)/2.
This proves that this sequence is limitlessly increasing toinfinity.

2. X_n = tan(−πn/[2(n+1)])
Let's prove that this sequence is limitlessly decreasing tonegative infinity.
Choose any boundary numberA, negative and however large by absolute value.
Let's find N for a chosen boundary - number A - by solving the inequality
tan(−πn/[2(n+1)]) ≤ A
which is equivalent to (sincetan(−φ) = −tan(φ))
−tan(πn/[2(n+1)]) ≤ A
or
tan(πn/[2(n+1)]) ≥ −A
Here −A is a positive number, however large, function tan is monotonically increasing on interval [0,π/2).
Expression πn/[2(n+1)] is monotonically increasing to π/2and its tangent is monotonically increasing to infinity as n is increasing. So, all we have to find is such n that
tan(πn/[2(n+1)]) = −A,
which happens at
πn/[2(n+1)] = arctan(−A)
or
πn = 2(n+1)arctan(−A)
From the above equation follows:
n(π−2arctan(−A)) =
= 2arctan(−A)
Therefore,
n = 2arctan(−A) /
/ (π−2arctan(−A))
Since n is a natural number, we have to choose the next natural number greater than the above expression.

3. X_n = n·sin(n)
This sequence is growing by absolute value, but changing the sign from positive to negative and back with the periodicity of2π
That means, the sequence cannot be qualified as having aninfinity (positive or negative) as a limit.
It's not infinitely growing topositive infinity, nor infinitely decreasing to negative infinity.