Monday, August 22, 2016

Unizor - Derivatives - Limit of Ratio of Polynomials

Notes to a video lecture on

Sequence Limit -
Ratio of Polynomials

When a sequence is represented by a ratio of two polynomials of order number n, it's easy to find its limit.
It's all about the members of the highest power in numerator and denominator.

Assume, a sequence is given by an expression
Xn = P(n) / Q(n)
where P(n) is a polynomial of power p (here p - some natural number) of n and Q(n) is a polynomial of power q (here q - some natural number) of n.

So, we can write the following expressions for our polynomials:
P(n) = a0np+a1np−1+...+apn0
(where a0 ≠ 0)
Q(n) = b0nq+b1nq−1+...+bqn0
(where b0 ≠ 0)

In order to determine the limit of their ratio, let's transform them as follows:

Consider two expressions in parenthesis:
R(n) = a0+a1n−1+...+apn−p
S(n) = b0+b1n−1+...+bpn−q

As order number n increases to infinity, each member of these expressions, except the first (aand b0) is an infinitesimal and, therefore, has its limit equal to0. Since there is only finite number of these infinitesimals in each expression, the limit of the first one is a0 and the limit of the second one is b0.
That means that the limit of their ratio is a0 / b0, that is
R(n) / S(n) → a0 / b0.

Let's return back to our original ratio of two polynomials.
P(n) / Q(n) =
= [np·R(n)] / [nq·S(n)] =
= np−q·[R(n) / S(n)]

Ratio [R(n) / S(n)] has a limit a0 / b0 and, therefore, is bounded.
Expression np−q is either
infinitesimal (for p less than q) or
constant 1 (for p equaled to q) or
infinitely growing (for p greater than q)

Therefore, the ratio of original polynomials P(n) / Q(n) is a sequence that
(a) is infinitesimal
if p is less than q
(b) converges to a limit a0 / b0
if p equals to q
(c) is infinitely growing
if p is greater than q

We have reduced a problem for the limit of a ratio of two polynomials to a simple comparing their highest powers (p and q) and corresponding coefficients at the members in these powers (a0 and b0).

We can easily expand this approach to a ratio of two functions that can be represented as a sum of finite number of power functions, like
Xn = U(n) / V(n)
U(n) = Σui·npi (0 ≤ i ≤ M)
V(n) = Σvi·nqj (0 ≤ j ≤ N)

In the above expressions powers pi and qj can be any real numbers, not necessarily natural (like 0.5 for a square root or even irrational powers like π).

All we have to do now is position all members of U(n) in order of decreasing of their powers, same with V(n), and factor out that highest power. Remaining members (analogous to R(n) and S(n)above) will contain only members with negative powers, except the first constants (uand v0 correspondingly), and they all will converge.

Therefore, assuming our functions U(n) and V(n) are already written in the order of decreasing powers of their members, the following expression would correctly represent our ratio
U(n) / V(n) = np0−q0·W(n)
where p0 is the highest power among members of U(n)q0 is the highest power among members of V(n)W(n) is a sequence converging to u0 / v0- a ratio of coefficients at the highest powers of functions U(n) and V(n).

Hence, we can make a judgment about convergence of our ratio U(n) / V(n).
(a) it is infinitesimal
if p0 (the highest power of U(n)) is less than q0 (the highest power of V(n))
(b) converges to a limit a0 / b0
if p0 equals to q0
(c) is infinitely growing
if p0 is greater than q0.

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