Limits of Roots

From Limits of Powers to Limits of Roots

The previous tutorial established that a fixed positive integer power preserves limits: if \(a_n\to A\), then \(a_n^m\to A^m\). Roots reverse this operation, but their limit behavior requires attention to their domains. Even roots are real only for nonnegative inputs, while odd roots are defined for inputs of either sign.

Throughout this tutorial, the root index \(m\) is a fixed positive integer. We will show that taking the corresponding root preserves convergence whenever the terms lie in the root's domain. The key step is an inequality: for nonnegative numbers, the \(m\)-th power of the difference between two numbers cannot exceed the absolute difference between their \(m\)-th powers.

Real Roots and a Useful Inequality

For a positive integer \(m\), the \(m\)-th root of \(x\geq0\) is the unique nonnegative real number \(r\) such that \(r^m=x\). When \(m\) is odd, the \(m\)-th root is defined for every real \(x\): it is the unique real number \(r\) such that \(r^m=x\). We write this root as \(\sqrt[m]{x}\). For even \(m\), the notation \(\sqrt[m]{x}\) refers to the nonnegative root.

The following inequality will turn convergence of the radicands into convergence of their roots.

Proof. If \(s\geq t\), write \(s=t+d\), where \(d=s-t\geq0\). The binomial theorem gives

Every term in this sum is nonnegative, and the term with \(j=m\) is \(d^m\). Therefore \(s^m-t^m\geq d^m=|s-t|^m\). If \(t\geq s\), interchange \(s\) and \(t\) in this argument to obtain \(t^m-s^m\geq|s-t|^m\). In either case, \(|s^m-t^m|\geq|s-t|^m\), as required. \(\square\)

In particular, if \(s\) and \(t\) are nonnegative \(m\)-th roots of \(x\) and \(y\), then \(s^m=x\) and \(t^m=y\). The lemma becomes

This estimate is useful even when \(s\) and \(t\) are close to zero. It does not require dividing by a root or assuming that the limit of the radicands is positive.

The Limit Theorem for Roots

Proof. First suppose \(m\) is even. The assumption \(a_n\geq0\) for every \(n\), together with the Theorem (Order Preservation for Limits), implies \(A\geq0\). Set \(u_n=\sqrt[m]{a_n}\) and \(u=\sqrt[m]{A}\). Both \(u_n\) and \(u\) are nonnegative, and \(u_n^m=a_n\), \(u^m=A\). The Difference Inequality for Nonnegative Powers gives

Let \(\varepsilon>0\). Since \(\varepsilon^m>0\) and \(a_n\to A\), there is an \(N\) such that \(|a_n-A|<\varepsilon^m\) whenever \(n\geq N\). For those \(n\),

Thus \(u_n\to u\), proving the even-root case.

Now suppose \(m\) is odd. If \(A=0\), then \(|\sqrt[m]{a_n}|^m=|a_n|\). Given \(\varepsilon>0\), convergence \(a_n\to0\) gives an \(N\) such that \(|a_n|<\varepsilon^m\) for \(n\geq N\). Hence \(|\sqrt[m]{a_n}|<\varepsilon\), which proves \(\sqrt[m]{a_n}\to0=\sqrt[m]{A}\).

It remains to consider \(A\neq0\). By the Theorem (Eventual Sign Stability), \(a_n\) has the same sign as \(A\) for all sufficiently large \(n\). Odd roots have the same sign as their radicands, so \(\sqrt[m]{a_n}\) and \(\sqrt[m]{A}\) also have the same sign for those indices. Their absolute values are the nonnegative \(m\)-th roots of \(|a_n|\) and \(|A|\). Applying the Difference Inequality to these absolute values gives

The equality uses the fact that the roots have the same sign, and the last inequality is the Reverse Triangle Inequality. Given \(\varepsilon>0\), convergence \(a_n\to A\) makes the final quantity less than \(\varepsilon^m\) for all sufficiently large \(n\). The displayed estimate then gives \(|\sqrt[m]{a_n}-\sqrt[m]{A}|<\varepsilon\). This proves the odd-root case. \(\square\)

Worked Examples

When Root Convergence and Radicand Convergence Agree

The root limit theorem has a converse. If a sequence of fixed roots converges, then the sequence of radicands converges as well. This follows from the Limit of a Fixed Positive Integer Power Theorem established in the previous tutorial.

Proof. If \(a_n\to A\), then for even \(m\) the terms are nonnegative by the domain condition, and \(A\geq0\) by the Theorem (Order Preservation for Limits). For odd \(m\) there is no sign restriction. In either case, the Limit of a Fixed Root Theorem shows that \(\sqrt[m]{a_n}\) converges.

Conversely, suppose \(\sqrt[m]{a_n}\to R\). Set \(r_n=\sqrt[m]{a_n}\). By the definition of an \(m\)-th root, \(r_n^m=a_n\) for every \(n\). Since \(m\) is a fixed positive integer, the Limit of a Fixed Positive Integer Power Theorem gives \(r_n^m\to R^m\). Thus \(a_n\to R^m\), proving the converse and the stated limit. \(\square\)

This criterion concerns a fixed root index. It cannot be used without further argument when the index varies with \(n\). For instance, the sequence \(\sqrt[n]{a_n}\) does not apply the same function to every term; its root index changes, just as a changing exponent falls outside the fixed-power theorem. First check whether the root index is fixed and whether every term belongs to the root's domain.

Domain Conditions and Common Pitfalls

For even roots, requiring \(a_n\geq0\) at every index ensures that the sequence of roots is defined as a real sequence. If those terms converge, their limit cannot be negative, by the Order Preservation for Limits theorem. It is not enough for the radicands to be nonnegative only eventually if the expression is meant to define a real sequence at every index.

Odd roots have a different domain: negative inputs are allowed. The proof also shows why the zero limit deserves a separate case. When \(A=0\), smallness of \(|a_n|\) directly controls the magnitude of its odd root. When \(A\neq0\), eventual sign stability lets us compare the roots through their nonnegative magnitudes.

Finally, the root index must remain fixed. The estimate in the proof uses the same exponent \(m\) throughout and chooses the input tolerance \(\varepsilon^m\). If \(m\) changes with \(n\), this argument does not establish convergence. The fixed-root theorem and the changing-index problem are distinct questions.