Calculus II: Sequences and Convergence

Sequences are the fundamental building blocks of many advanced concepts in calculus, engineering, and computer science. Understanding when and how a sequence settles toward a specific value—or fails to do so—is critical for analyzing everything from alternating current signals to the long-term behavior of iterative algorithms. This unit moves beyond the mechanics of series to establish the rigorous logic of convergence, the property that underpins the very definition of an infinite sum.

Defining Sequences and Their Limits

A sequence is an ordered list of numbers, typically written as $a_1,a_2,a_3,...$ or $\{a_n{\}}_{n=1}^{\infty }$. You can define it with an explicit formula, where the $n$th term is a direct function of $n$, such as $a_n=\frac{n}{n+1}$. Alternatively, you can use a recursive definition, where each term depends on previous ones, like the Fibonacci sequence: $F_1=1,F_2=1,F_n=F_{n-1}+F_{n-2}$.

The central question is the sequence's long-term behavior. We say a sequence $\{a_n\}$ converges to a limit $L$ if, as $n$ becomes arbitrarily large, the terms $a_n$ get and stay arbitrarily close to $L$. Formally, for every $ϵ>0$ (a tiny distance), there exists an integer $N$ such that for all $n>N$, we have $|a_n-L|<ϵ$. If no such $L$ exists, the sequence diverges. For the example $a_n=\frac{n}{n+1}$, the limit is 1, written ${\lim }_{n\to \infty }{a}_n=1$.

Tools for Determining Convergence: The Squeeze Theorem and Boundedness

You won't always calculate limits directly. The Squeeze Theorem for sequences is a powerful indirect tool. If you can find two convergent sequences, $\{b_n\}$ and $\{c_n\}$, that "trap" your sequence such that $b_n\le a_n\le c_n$ for all sufficiently large $n$, and ${\lim }_{n\to \infty }{b}_n={\lim }_{n\to \infty }{c}_n=L$, then your sequence is forced to converge to the same limit: ${\lim }_{n\to \infty }{a}_n=L$. This is particularly useful for sequences involving oscillatory factors like $(-1{)}^n$. For instance, since $-1/n\le \frac{(-1{)}^n}{n}\le 1/n$ and both bounding sequences go to 0, the squeezed sequence also converges to 0.

Relatedly, a sequence is bounded if there exists a number $M$ such that $|a_n|\le M$ for all $n$. While a convergent sequence is always bounded, the converse is not true: a bounded sequence may not converge (e.g., $a_n=(-1{)}^n$ is bounded by 1 but diverges). Boundedness is a necessary but not sufficient condition for convergence.

The Monotone Convergence Theorem

To get a sufficient condition, we combine boundedness with monotonicity. A sequence is monotonic if it is either entirely non-decreasing ($a_{n+1}\ge a_n$) or non-increasing ($a_{n+1}\le a_n$). The Monotone Convergence Theorem (MCT) is a cornerstone result: a sequence that is both bounded and monotonic must converge. This theorem is incredibly valuable because it guarantees convergence without requiring you to know the actual limit.

For example, consider a recursively defined sequence: $a_1=2,a_{n+1}=\frac{1}{2}(a_n+\frac{6}{a_n})$, which approximates $\sqrt{6}$. You can first show it is bounded below (e.g., by $\sqrt{6}$) and then prove it is decreasing. By the MCT, it must converge to some limit $L$. You can then find $L$ by taking the limit of both sides of the recursive rule: $L=\frac{1}{2}(L+\frac{6}{L})$, which solves to $L=\sqrt{6}$.

Comparing Growth Rates and Divergence

When sequences diverge, they often do so by growing without bound. Comparing growth rates of different functions of $n$ helps you quickly determine limits at infinity. For $n\to \infty$, the hierarchy of growth is generally: logarithms < polynomials < exponential functions. Knowing this, you can instantly reason that ${\lim }_{n\to \infty }\frac{\ln n}{n^2}=0$ and ${\lim }_{n\to \infty }\frac{3^n}{n^{100}}=\infty$.

A critical application is the nth-Term Test for Divergence of a series. It states: if the limit of the sequence of terms is not zero (${\lim }_{n\to \infty }{a}_n\ne 0$), then the corresponding infinite series ${\sum }_{n=1}^{\infty }{a}_n$ must diverge. This is a direct link from sequence convergence to series convergence. Crucially, the converse is false: if ${\lim }_{n\to \infty }{a}_n=0$, the series may converge or diverge (e.g., the harmonic series diverges despite its terms going to zero). This distinction is a common conceptual hurdle.

Common Pitfalls

1. Assuming Sequence Limit Implies Series Convergence: This is the most frequent error. Remember: ${\lim }_{n\to \infty }{a}_n=0$ is a necessary condition for the series $\sum a_n$ to converge, but it is not sufficient. The nth-Term Test can only prove divergence; it can never prove convergence.
2. Misapplying L'Hôpital's Rule: L'Hôpital's Rule applies to functions, not directly to sequences. To use it on a sequence limit like ${\lim }_{n\to \infty }\frac{\ln n}{n}$, you must first consider the corresponding function limit ${\lim }_{x\to \infty }\frac{\ln x}{x}$.
3. Confusing Recursive and Explicit Analysis: When working with a recursively defined sequence, you cannot find its limit by simply "plugging in infinity" into the recursive formula. You must first use tools like the Monotone Convergence Theorem to prove convergence exists, then use the limit properties on the recursive equation to solve for the limit $L$.
4. Overlooking Boundedness in the MCT: The Monotone Convergence Theorem requires both properties. A monotonic sequence that is not bounded (e.g., $a_n=n$) diverges to infinity. A bounded sequence that is not monotonic (e.g., $(-1{)}^n$) may diverge.

Summary

• A sequence $\{a_n\}$ is a list of numbers, defined explicitly or recursively. It converges to limit $L$ if its terms become and remain arbitrarily close to $L$ as $n$ increases.
• The Squeeze Theorem and the Monotone Convergence Theorem are key tools for proving convergence, often when a direct limit calculation is difficult.
• Understanding growth rate comparisons (logs vs. polynomials vs. exponentials) allows for rapid evaluation of many sequence limits as $n\to \infty$.
• The convergence of the sequence of terms is fundamentally linked to the convergence of the corresponding infinite series via the nth-Term Test for Divergence: if ${\lim }_{n\to \infty }{a}_n\ne 0$, then $\sum a_n$ diverges.
• The logic of the nth-Term Test is one-way: a limit of zero does not guarantee series convergence, making it a divergence test only.