Unizor - Derivatives - Infinitesimals

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


Sequence Limit - Infinitesimal
(infinitely small)

Special role in mathematics in general and, in particular, in calculus is played by sequences that converge to zero.
These sequences have a special name - infinitesimal. More descriptive name might beinfinitely small.

It is very important to understand that here we are not dealing with any concrete, however small, number, but with a sequence converging to zero.

When we say that ε is an infinitesimal value, we mean that it represents a sequence of values converging to zero, that is a process, a variable that changes its value, gradually getting closer and closer to zero.

When we say that a distance between two objects is an infinitesimal, we imply that these objects are moving towards each other such that the distance between them converges to zero.

When we say that the speed of an object changes by an infinitesimal value during infinitesimal time interval, we mean the following process:
(a) we fix some moment in timeT₀ and the speed of an object at this moment V₀;
(b) we consider an infinite sequence of time intervals starting at T₀ and ending at T₁,T₂,...T_n... such that the difference in time |T_n − T₀|converges to zero as index nincreases;
(c) we measure a speed V_n of an object at each end of intervalT_n;
(d) the difference between the original speed V₀ and the speed at each end of interval V_n is a sequence that converges to zero:
|V_n − V₀| → 0

Properties of infinitesimals are direct consequences of properties of the limits in general:

1. If ε is an infinitesimal (more precisely, if a sequence {ε_n}converges to zero, but we will use the former expression for brevity), then K·ε is also an infinitesimal, where K - any real constant, positive, negative or zero.

2. If ε and δ are two infinitesimals, their sum ε+δ is an infinitesimal as well.

3. If ε and δ are two infinitesimals, their product ε·δis an infinitesimal as well.

Lots of problems are related to a division of one infinitesimal by another, provided the infinitesimal in the denominator does not take the value of zero. The result of this operation can be a sequence that might or might not converge at all and, if it converges, it can converge to any number.
Here are a few examples.

A) ε = {13/n}→0;
δ = {37/n}→0;
⇒ ε/δ = {13n/37n} = {13/37}.
Since sequence ε/δ is a constant, its limit is the same constant, that is 13/37.
Obviously, we can similarly construct two sequences with ratio converging to any number.

B) ε = {13/n}→0;
δ = {37n/(n²+1)}→0;
⇒ ε/δ = {[13(n²+1)]/37n²} =
= {(13/37)·[1+1/(n²+1)]} =
= {(13/37)+13/[37(n²+1)]} =
= 13/37 + γ → 13/37,
since γ is an infinitesimal {13/[37(n²+1)]}→0.

C) ε = {13/n²}→0;
δ = {37n/(n²+1)}→0;
⇒ ε/δ = {[13(n²+1)]/37n³} =
= {(13/37)·[1/n+1/(n²+1)]} =
= {13/(37n)+13/[37(n²+1)]} =
= γ₁ + γ₂ → 0,
since both γ₁=13/(37n) andγ₂=13/[37(n²+1)] are infinitesimals.

D) ε = {13/n}→0;
δ = {37/(n²+1)}→0;
⇒ ε/δ = {[13(n²+1)]/(37n)} =
= (13n)/37+13/(37n)
The last expression is growing limitlessly as n is increasing.
So, the result is not a convergent sequence. However, it is reasonable to say that this sequence "grows to positive infinity" as n increases.
We will discuss a concept of infinity later.

E) ε = {1/n}→0;
δ = {(−1)ⁿ/n}→0;
⇒ ε/δ = {(−1)ⁿ}
The last expression represents a sequence that alternatively takes only two values, 1 and −1. It is not converging to any number, nor can we say that it grows to infinity.

As we see, a ratio of two infinitesimals might be convergent or not-convergent sequence, bounded or unbounded.
If we suspect that it should converge to some limit, we have to resolve this by some transformation into an expression without division between two infinitesimals.

It should be noted, however, that the most interesting cases in dealing with infinitesimals occur exactly at the point of division of one by another. Numerous examples of this can be found in physics (like a concept of speed, where we divide an infinitesimal distance by infinitesimal time interval during which this distance is covered), analysis of smooth functions (like in determining their local maximums and minimums) and many others.