Polynomial Sequence Limits

Why the Highest Power Matters

When a polynomial is evaluated at increasingly large positive integers, its terms do not all contribute equally to the result. A term with a higher power of \(n\) eventually outweighs terms with lower powers. This observation lets us determine the long-term behavior of a polynomial sequence without calculating a long list of terms.

The leading term determines whether the values grow positively or negatively. The lower-degree terms still affect individual values, but we can prove that their combined size eventually becomes small compared with the leading term.

Polynomial Sequences and Their Limits

If \(d=0\), the polynomial is constant. If \(d\ge 1\), the leading term is \(a_dn^d\). The sign of \(a_d\) will determine the direction in which the sequence grows.

For clarity, \(\lim_{n\to\infty}a_n=+\infty\) means that for every real number \(M>0\), there is an index \(N\in\mathbb{N}\) such that \(a_n>M\) whenever \(n\ge N\). The definition of \(\lim_{n\to\infty}a_n=-\infty\) is the same with \(a_n<-M\). These statements describe unbounded growth, not convergence to a real number.

Worked Example: A Positive Leading Coefficient

Worked Example: A Negative Leading Coefficient

Why the Lower-Degree Terms Become Negligible

The examples used explicit bounds. The same idea proves a general result for every polynomial. Write a nonconstant polynomial as \(P(n)=a_dn^d+R(n)\), where \(d\ge1\) and \(R(n)=\sum_{k=0}^{d-1}a_kn^k\). Set \(C=\sum_{k=0}^{d-1}|a_k|\). For every \(n\ge1\), each power \(n^k\) in the remainder is at most \(n^{d-1}\), so

This proof explains the word “negligible” precisely. It does not claim that the lower-degree terms are zero. It shows that their total size, compared with \(|a_d|n^d\), can be made as small as desired. It also shows why the leading coefficient’s sign matters: eventually the remainder cannot reverse the sign of the leading term’s contribution.

Worked Example: A Constant Polynomial After Simplification

Worked Example: A Linear Polynomial

Common Pitfall: A Normalized Limit Is Not the Original Limit

Dividing a polynomial by its highest power can reveal its leading coefficient, but the resulting limit belongs to the normalized expression. It does not give the limit of the original polynomial.

For example, consider \(P(n)=2n^2+n\). Dividing by \(n^2\) gives

The normalized expression tends to \(2\). A mistaken conclusion would be: “Therefore \(P(n)\to2\).” That conclusion changes the expression whose limit is being considered. In fact, for \(n\ge1\), \(P(n)=2n^2+n\ge2n^2\), so for any \(M>0\), choosing an integer \(N>\sqrt{M/2}\) gives \(P(n)>M\) whenever \(n\ge N\). Thus \(P(n)\to+\infty\), not \(2\).

The leading coefficient describes the polynomial after division by its leading power. The original polynomial has that leading term multiplied by \(n^d\), so its behavior must be determined from the full expression.

Summary

For a nonconstant polynomial sequence \(P(n)\), first combine like powers and identify the degree \(d\) and leading coefficient \(a_d\). The lower-degree terms have total size at most \(Cn^{d-1}\), while the leading term has size \(|a_d|n^d\). Their ratio is bounded by \(C/(|a_d|n)\), which tends to zero. Consequently, a positive leading coefficient gives divergence to \(+\infty\), and a negative leading coefficient gives divergence to \(-\infty\). A constant polynomial instead gives a constant sequence and has that constant as its finite limit.