Unizor - Derivatives - Limit of Ratio of Polynomials

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


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
X_n = 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) = a₀n^p+a₁n^p−1+...+a_pn⁰
(where a₀ ≠ 0)
Q(n) = b₀n^q+b₁n^q−1+...+b_qn⁰
(where b₀ ≠ 0)

In order to determine the limit of their ratio, let's transform them as follows:
P(n)=n^p(a₀+a₁n⁻¹+...+a_pn^−p)
Q(n)=n^q(b₀+b₁n⁻¹+...+b_pn^−q)

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

As order number n increases to infinity, each member of these expressions, except the first (a₀ and 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) =
= [n^p·R(n)] / [n^q·S(n)] =
= n^p−q·[R(n) / S(n)]

Ratio [R(n) / S(n)] has a limit a₀ / b₀ and, therefore, is bounded.
Expression n^p−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
X_n = U(n) / V(n)
where
U(n) = Σu_i·n^p_i (0 ≤ i ≤ M)
and
V(n) = Σv_i·n^q_j (0 ≤ j ≤ N)

In the above expressions powers p_i and q_j 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 (u₀ and 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) = n^p₀−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.