Why Small Angles Simplify Trigonometric Terms
In a sequence, a trigonometric expression can look complicated even when its angle becomes very small. The key observation is that near zero, sine and tangent behave like the angle itself, while one minus cosine behaves like half the square of the angle. These facts turn many trigonometric terms into familiar power-law comparison models.
For example, if an angle is \(3/n\), it approaches zero as \(n\) grows. Thus \(\sin(3/n)\) has the same leading behavior as \(3/n\). But \(1-\cos(3/n)\) has leading behavior \(9/(2n^2)\), not \(3/n\). Distinguishing these orders is essential when comparing positive-term series.
Throughout, angles are measured in radians. The standard small-angle limits below depend on this convention.
Asymptotic Equivalence and the Small-Angle Rules
For sequences \(a_n\) and \(b_n\), with \(b_n\ne 0\) for all sufficiently large \(n\), the notation \(a_n\sim b_n\) means that their ratio tends to one. In particular, the two sequences have the same leading behavior, though they need not be equal.
The standard small-angle limits are first stated as limits of a real variable \(x\) approaching zero.
Equivalently, as \(x\to 0\), \(\sin x\sim x\), \(\tan x\sim x\), and \(1-\cos x\sim x^2/2\). When using these facts in a sequence, substitute its angle for \(x\), provided that angle tends to zero and the expressions are defined.
Why the Three Equivalents Hold
The sine limit is the basic small-angle limit. One standard justification uses the inequalities \(\cos x\le \sin x/x\le 1\) for \(0<x<\pi/2\). As \(x\to0^+\), both outer expressions tend to one, so the squeeze theorem gives \(\sin x/x\to1\). Since \(\sin x/x\) is even, the same limit holds from the left.
The tangent limit follows from the definitions and the sine limit. For \(x\) near zero with \(x\ne0\), \(\tan x=\sin x/\cos x\), and \(\cos x\to1\). Therefore
To obtain the cosine equivalent, use the identity \(1-\cos x=2\sin^2(x/2)\). Then
This also explains the different order: the sine of a small angle is approximately linear in that angle, but one minus cosine is approximately quadratic.
Applying the Limits to Sequences
Suppose an angle \(u_n\) tends to zero. The small-angle rule replaces a trigonometric expression by a simpler expression in \(u_n\). If \(u_n\) itself is built from \(n\), simplify that resulting expression to identify the power of \(n\) and its constant factor. For example, \(u_n=5/n^2\) gives \(\sin(u_n)\sim 5/n^2\), whereas \(1-\cos(u_n)\sim 25/(2n^4)\).
For positive-term series, an equivalent comparison is useful because a standard limit-comparison result says that if \(a_n,b_n>0\) eventually and \(a_n/b_n\to L\) for some finite \(L>0\), then \(\sum a_n\) and \(\sum b_n\) have the same convergence behavior. Asymptotic equivalence is the special case \(L=1\). It is enough to compare all sufficiently large terms; changing or omitting finitely many terms does not affect convergence.
Worked Example: A Sine Term With a Rational Factor
Find an equivalent for \(a_n=\sin(3/n)/(n+2)\), and determine whether \(\sum_{n=1}^{\infty}a_n\) converges.
First, \(3/n\to0\), and the sine term is positive because \(0<3/n\le3<\pi\). The small-angle limit gives \(\sin(3/n)\sim3/n\). Including the denominator suggests the comparison term \(b_n=3/n^2\). Verify the ratio directly:
Thus \(a_n\sim3/n^2\). The comparison series \(\sum 3/n^2\) converges, since it is a constant multiple of a \(p\)-series with \(p=2>1\). The limit comparison therefore shows that \(\sum a_n\) converges.
Check: The ratio of the original term to the proposed equivalent tends to one, so the factor \(3\) and the power \(n^{-2}\) are both accounted for.
Worked Example: A Tangent Term With a Square-Root Angle
For \(n\ge2\), let \(a_n=\tan(2/\sqrt n)/(n+1)\). Find an equivalent and decide whether its series converges. Starting at \(n=2\) ensures that the angle lies between zero and \(\pi/2\), so the tangent is defined and positive.
The angle \(2/\sqrt n\) tends to zero. Since \(\tan u\sim u\), the numerator is equivalent to \(2/\sqrt n\). Including the denominator suggests \(b_n=2/n^{3/2}\). The ratio confirms this:
Therefore \(a_n\sim2/n^{3/2}\). The series \(\sum 2/n^{3/2}\) converges because its \(p\)-series exponent is \(3/2>1\). By limit comparison, \(\sum_{n=2}^{\infty}a_n\) converges as well.
Check: The denominator contributes a factor of approximately \(n^{-1}\), and the tangent contributes \(2n^{-1/2}\). Their product has order \(2n^{-3/2}\), matching the ratio calculation.
Worked Example: One Minus Cosine at a Square-Root Angle
Let \(c_n=1-\cos(4/\sqrt n)\). Find its equivalent and determine the behavior of \(\sum_{n=1}^{\infty}c_n\).
The angle \(4/\sqrt n\) tends to zero. The cosine rule gives \(1-\cos u\sim u^2/2\). Substitution yields the candidate \(b_n=(4/\sqrt n)^2/2=8/n\). Check by taking the ratio:
Thus \(c_n\sim8/n\). The comparison series \(\sum 8/n\) diverges, since it is a positive constant multiple of the harmonic series. The terms \(c_n\) are positive, and limit comparison shows that \(\sum c_n\) diverges.
Check: Squaring the angle \(4/\sqrt n\) gives \(16/n\), and dividing by two gives \(8/n\). This confirms both the power and the constant in the equivalent.
A Common Mistake: Treating One Minus Cosine as Linear
A frequent error is to assume that \(1-\cos x\sim x\), perhaps because both expressions approach zero. But two expressions tending to zero need not be equivalent. Equivalence requires their ratio to tend to one.
For \(x\ne0\), the actual ratio for this incorrect proposal is
not one. The proposed linear equivalent is therefore wrong. For a concrete series, consider \(1-\cos(1/n)\). The incorrect rule would suggest \(1/n\), but the correct equivalent is
The harmonic comparison suggested by the wrong rule would predict divergence; the correct \(p\)-series comparison shows convergence. Always square the small angle when applying the one-minus-cosine rule.
Choosing and Checking a Comparison
Small-angle equivalents are most useful when the angle is written clearly before any approximation is made. Identify the angle \(u_n\), verify that \(u_n\to0\), and only then replace the trigonometric expression. Keep constants as well as powers: a positive constant does not change the convergence behavior of a comparison series, but retaining it helps verify that the equivalent is correct.
| Term | Equivalent when \(u_n\to0\) | Condition to check |
|---|---|---|
| \(\sin(u_n)\) | \(u_n\) | The angle tends to zero; for positive terms, \(u_n\) is positive eventually. |
| \(\tan(u_n)\) | \(u_n\) | The angle tends to zero and the tangent is defined; for positive terms, \(u_n\) is positive eventually. |
| \(1-\cos(u_n)\) | \(u_n^2/2\) | The angle tends to zero and is nonzero eventually. |
After proposing an equivalent \(b_n\), a direct check of \(a_n/b_n\to1\) is a reliable way to catch a missing constant or an incorrect power. If the terms are positive eventually, that equivalence then transfers the convergence behavior of a known comparison series.
Check Your Understanding
For each question, state the relevant small-angle rule and show how you would check your proposed equivalent.
- Find an equivalent for \(\sin(5/n^2)\).
- Find an equivalent for \(\tan(3/\sqrt n)/n^2\).
- Find an equivalent for \(1-\cos(2/n^{1/3})\).
- Why does \(\sin(u_n)\sim u_n\) require checking positivity separately when applying a positive-term series comparison?
- Explain why \(1-\cos x\sim x\) is false by evaluating the ratio of the two expressions as \(x\to0\).