Eventually Positive Sequences

Positivity After a Threshold

The previous tutorial described sequences whose terms eventually pass every real bound. Eventual positivity is a more modest condition: instead of requiring terms to grow without bound, we ask only that they eventually stay above zero. The sequence may be bounded, may converge, or may continue to vary. What matters is that one index marks a point after which every term is positive.

As with divergence to infinity, the word “eventually” allows finitely many exceptions. Those early terms can be negative or zero without affecting whether the sequence is eventually positive. The threshold must, however, work for every later index, not just for infinitely many positive terms.

Every eventually positive sequence is eventually nonnegative, but the converse need not hold: a sequence that is zero from some point onward is eventually nonnegative without being eventually positive. Keeping the inequalities strict or non-strict is important, especially when using positivity in a product or when discussing whether a limit can equal zero.

Finding a Threshold

Basic Operations Preserve Eventual Positivity

Once each of two sequences is positive beyond some index, their sum and product are positive beyond a common index. The two original thresholds might differ, so we use the larger one. This is a basic but useful way to handle expressions assembled from several sequences.

Proof. Since \((a_n)\) is eventually positive, there is an \(N_a\in\mathbb{N}_0\) such that \(a_n>0\) for every \(n\geq N_a\). There is also an \(N_b\in\mathbb{N}_0\) such that \(b_n>0\) for every \(n\geq N_b\). Let \(N=\max\{N_a,N_b\}\). For every \(n\geq N\), both inequalities hold, so \(a_n+b_n>0\) and \(a_nb_n>0\). This proves the sum and product assertions. If \(c>0\) and \(a_n>0\) for every \(n\geq N_a\), then \(ca_n>0\) for every \(n\geq N_a\). Hence \((ca_n)\) is eventually positive as well. \(\square\)

The requirement \(c>0\) matters for the scalar-multiple assertion: multiplication by a negative constant reverses the sign. If \(c=0\), the resulting sequence is zero and is not eventually positive. Similarly, the sum of an eventually positive sequence and an eventually nonnegative sequence is eventually positive: beyond the larger of their thresholds, it is the sum of a positive number and a nonnegative number.

Eventual Comparison

An inequality can transfer eventual positivity from one sequence to another. The comparison only needs to hold on a tail, and the sequence being compared against must be at least as large as the positive sequence there.

Proof. Since \((a_n)\) is eventually positive, choose \(N_1\in\mathbb{N}_0\) such that \(a_n>0\) for every \(n\geq N_1\). Let \(N=\max\{N_0,N_1\}\). For every \(n\geq N\), the comparison and positivity both apply, giving

Thus \(b_n>0\) for every \(n\geq N\), which is exactly the definition of eventual positivity. \(\square\)

The direction of comparison cannot be reversed without further information. A sequence lying below an eventually positive sequence could still be negative. For example, \(a_n=1\) and \(b_n=-1\) satisfy \(b_n\leq a_n\) at every index, but \(b_n\) is not eventually positive. The theorem uses the lower sequence as the positive one and the upper sequence as the one whose positivity is concluded.

What Eventual Positivity Says About a Limit

Eventual positivity constrains the limit of a convergent sequence, but it does not force that limit to be strictly positive. The following result rules out negative limits. Its proof makes the contradiction explicit: convergence to a negative number would eventually place the terms below zero, in conflict with eventual positivity.

Proof. Suppose, to the contrary, that \(L<0\). Then \(\varepsilon=-L/2\) is positive. Since \(a_n\to L\), there is an \(N_1\in\mathbb{N}_0\) such that \(|a_n-L|<-L/2\) whenever \(n\geq N_1\). In particular,

for every \(n\geq N_1\). But eventual positivity gives an \(N_2\in\mathbb{N}_0\) such that \(a_n>0\) for every \(n\geq N_2\). For any \(n\geq\max\{N_1,N_2\}\), the same term would have to satisfy both \(a_n<0\) and \(a_n>0\), an impossibility. Therefore \(L\) cannot be negative, and \(L\geq0\). \(\square\)

There is a useful converse when the limit is strictly positive. By Eventual Sign Stability, if \(a_n\to L\) with \(L\neq0\), then the terms eventually have the sign of \(L\). In particular, if \(L>0\), the sequence is eventually positive. Together with the theorem just proved, this gives a precise statement: a convergent eventually positive sequence has a nonnegative limit, and a convergent sequence with a positive limit is eventually positive. The zero-limit case is different and allows either sign behavior.

Common Pitfalls

Eventual positivity is a statement about the sign of every term on a tail. It is not enough that positive terms occur infinitely often. For example, \(w_n=(-1)^n\) has positive terms at every even index, however large the index, but it also has negative terms at every odd index. No threshold makes all later terms positive.

It is also important not to confuse eventual positivity with a uniform positive lower bound. The sequence \(1/(n+2)\) is eventually positive, but given any \(c>0\), its terms eventually become smaller than \(c\). Eventual positivity only supplies the lower bound zero on a tail; it does not supply a positive number that all terms on that tail exceed.

Finally, finite initial exceptions are permitted, but infinitely many exceptions are not. When checking a candidate threshold, verify the required inequality for every index at or beyond it. If the argument combines two eventual conditions, take the larger of their thresholds so that both conditions hold simultaneously. This simple step is what allows the sum, product, and comparison results to apply without overlooking any indices.