Asymptotic Equivalence and Series Behavior

Why Compare Terms Asymptotically?

Some positive series have terms whose exact formulas are awkward, even though their long-term size resembles that of a familiar sequence. For instance, a term may contain a radical or a rational expression with several powers of \(n\). If its ratio to a simple comparison term approaches \(1\), the complicated expression can be replaced by that simpler term when deciding whether the series converges.

This replacement is about convergence behavior, not equality of terms or equality of sums. We will define the relationship precisely and prove why it preserves convergence for series with positive terms.

Definition of Asymptotic Equivalence

Let \((a_n)\) and \((b_n)\) be sequences that are positive for all sufficiently large \(n\). Their ratio is then defined for all sufficiently large \(n\).

The limit says that the ratio of the two terms becomes arbitrarily close to \(1\). Equivalently, for any chosen relative tolerance \(\varepsilon>0\), the terms eventually satisfy \(1-\varepsilon<a_n/b_n<1+\varepsilon\). In particular, choosing a fixed tolerance gives useful inequalities that compare the terms directly.

The Series Equivalence Theorem

The theorem applies to positive terms because the ratio limit yields inequalities between the terms, and the comparison test transfers convergence through such inequalities. Positivity also ensures that the partial sums increase, so a divergent positive-term series diverges to \(+\infty\).

Why the Theorem Works

Here is a complete proof using the definition of the limit. Since \(a_n/b_n\to1\), take the tolerance \(\varepsilon=1/2\). There is an index \(N\) such that whenever \(n\ge N\),

Removing the absolute value gives two inequalities:

Because \(b_n>0\), multiplying throughout by \(b_n\) preserves the inequality directions:

If \(\sum b_n\) converges, then the upper bound \(a_n<\frac32 b_n\) shows that the tail of \(\sum a_n\) converges by direct comparison. Adding the finite initial sum does not change convergence, so \(\sum a_n\) converges.

For the converse, rearrange the lower bound to obtain \(b_n<2a_n\) for \(n\ge N\). If \(\sum a_n\) converges, direct comparison shows that \(\sum b_n\) converges as well. Thus either both series converge or neither does. Since all terms are positive, if they do not converge, their partial sums are unbounded and tend to \(+\infty\). This proves the theorem.

The constants \(1/2\) and \(3/2\) are convenient choices, not special features of the result. Any two fixed positive bounds surrounding \(1\) would give the same comparison argument.

Applying the Theorem

What Asymptotic Equivalence Does—and Does Not—Say

The theorem allows a simpler sequence to stand in for a complicated one when the question is whether a positive-term series converges. It does not say that the terms are equal, that their partial sums are equal, or that convergent series have the same sum. For example, a finite number of terms can be changed without altering convergence, while those changes can alter the value of a convergent sum.

It is also important that the ratio limit be \(1\) for the notation \(a_n\sim b_n\). A finite positive ratio limit other than \(1\) can still be useful in a comparison argument, but it is not asymptotic equivalence under this definition. The key conclusion here is specifically that a ratio tending to \(1\) supplies eventual upper and lower bounds by fixed positive multiples.

A Common Mistake: Confusing Equivalence with a Vanishing Difference

A frequent error is to replace the ratio condition with the claim that the difference \(a_n-b_n\) tends to zero. That is not the definition. For example, let \(a_n=n+\sqrt n\) and \(b_n=n\). Then

so \(a_n\sim b_n\). But the difference is

not zero. The correction is to test the ratio \(a_n/b_n\), not the absolute difference. Asymptotic equivalence describes relative size: the difference can be large in absolute terms while remaining small compared with the terms themselves.

Another useful check is the direction of an inequality in the proof. From \(\frac12<a_n/b_n<\frac32\), positivity of \(b_n\) is what allows multiplication without reversing either inequality. If the terms are not eventually positive, this proof does not apply as written. The theorem is a result about positive-term series, not a general assertion about all series.

Practical Use

When a positive term has a complicated expression, a useful procedure is to identify a simpler positive comparison term, compute the ratio, and evaluate its limit. If the limit is \(1\), the two positive-term series have the same convergence behavior. The comparison series must still be one whose behavior is known; equivalence transfers the conclusion but does not determine it on its own.

The central idea is concise: for positive terms, a ratio approaching \(1\) eventually traps each sequence between fixed positive multiples of the other. Those inequalities are precisely what make the convergence behavior agree.