TRB BT Assistant - Convergence, Divergence and Cauchy Sequences E-Book Study Material

Convergence, Divergence and Cauchy Sequences

Introduction to Convergence of Sequences

In mathematics, a sequence is an ordered list of numbers. We often want to understand the behavior of these numbers as we go further and further down the list. The concept of convergence is central to understanding this behavior. A sequence is said to converge if its terms get arbitrarily close to a specific number, called the limit, as the index of the sequence increases indefinitely.

Formally, a sequence of real numbers $\{a_n\}_{n=1}^{\infty}$ converges to a real number $L$ if for every positive number $\epsilon$ (no matter how small), there exists a natural number $N$ such that for all $n > N$, the absolute difference between $a_n$ and $L$ is less than $\epsilon$. This is written as:

$\lim_{n \to \infty} a_n = L$

The $\epsilon$ here represents a tolerance. We are saying that after a certain point $N$ in the sequence, all subsequent terms $a_n$ must fall within the interval $(L - \epsilon, L + \epsilon)$.

Examples of Convergent Sequences

Let's look at some examples to solidify this understanding.

Sequence: $a_n = \frac{1}{n}$. For example: $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots$. As $n$ gets larger, $\frac{1}{n}$ gets closer and closer to 0. So, $\lim_{n \to \infty} \frac{1}{n} = 0$. For any $\epsilon > 0$, we can find an $N$ such that if $n > N$, then $\frac{1}{n} < \epsilon$. This $N$ can be chosen as $\lfloor \frac{1}{\epsilon} \rfloor + 1$.
Sequence: $a_n = c$, where $c$ is a constant. For example: $2, 2, 2, 2, \dots$. This sequence clearly converges to $c$. For any $\epsilon > 0$, we can choose any $N$ (e.g., $N=1$), and for all $n > N$, $|a_n - c| = |c - c| = 0 < \epsilon$.
Sequence: $a_n = \frac{n}{n+1}$. For example: $\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \dots$. As $n$ becomes very large, the '+1' in the denominator becomes insignificant, and the term approaches 1. So, $\lim_{n \to \infty} \frac{n}{n+1} = 1$. We can show this formally: $|\frac{n}{n+1} - 1| = |\frac{n - (n+1)}{n+1}| = |\frac{-1}{n+1}| = \frac{1}{n+1}$. For any $\epsilon > 0$, we need $\frac{1}{n+1} < \epsilon$, which means $n+1 > \frac{1}{\epsilon}$, or $n > \frac{1}{\epsilon} - 1$. So we can choose $N = \lfloor \frac{1}{\epsilon} - 1 \rfloor + 1$.

Divergence of Sequences

A sequence that does not converge is said to diverge. There are several ways a sequence can diverge:

Divergence to Infinity: A sequence $\{a_n\}$ diverges to positive infinity if for every positive number $M$, there exists a natural number $N$ such that for all $n > N$, $a_n > M$. This is written as $\lim_{n \to \infty} a_n = \infty$.
Divergence to Negative Infinity: A sequence $\{a_n\}$ diverges to negative infinity if for every negative number $M$ (or equivalently, for every positive number $M$, $a_n < -M$), there exists a natural number $N$ such that for all $n > N$, $a_n < M$. This is written as $\lim_{n \to \infty} a_n = -\infty$.
Oscillation: A sequence may oscillate between values and never settle down to a single limit.

Examples of Divergent Sequences

Let's look at examples of divergent sequences.

Sequence: $a_n = n$. For example: $1, 2, 3, 4, \dots$. This sequence grows without bound and diverges to positive infinity. For any $M > 0$, we can choose $N = \lfloor M \rfloor + 1$. Then for $n > N$, $a_n = n > N > M$.
Sequence: $a_n = (-1)^n$. For example: $-1, 1, -1, 1, -1, 1, \dots$. This sequence oscillates between -1 and 1. It does not approach any single limit $L$. If we claim $L=1$, then for $\epsilon = 0.5$, there is no $N$ such that for all $n > N$, $|(-1)^n - 1| < 0.5$. For odd $n$, $|-1 - 1| = 2$, which is not less than 0.5. If we claim $L=-1$, then for even $n$, $|1 - (-1)| = 2$, which is not less than 0.5.
Sequence: $a_n = n^2 - 1$. For example: $0, 3, 8, 15, \dots$. This sequence also diverges to positive infinity.
Sequence: $a_n = \sin(n)$. This sequence oscillates in a more complex way between -1 and 1 and does not converge.

Monotone Sequences

A sequence $\{a_n\}$ is called monotone if it is either entirely non-decreasing or entirely non-increasing.

Non-decreasing: $a_n \le a_{n+1}$ for all $n$.
Non-increasing: $a_n \ge a_{n+1}$ for all $n$.

A sequence is strictly increasing if $a_n < a_{n+1}$ for all $n$, and strictly decreasing if $a_n > a_{n+1}$ for all $n$. Monotone sequences have a very important property related to convergence.

The Monotone Convergence Theorem

This theorem is a cornerstone of real analysis. It states:

A non-decreasing sequence that is bounded above converges.
A non-increasing sequence that is bounded below converges.

If a monotone sequence is not bounded, it diverges to $\infty$ (if non-decreasing) or $-\infty$ (if non-increasing).

Example: Consider the sequence defined by $a_1 = 1$ and $a_{n+1} = \sqrt{2 + a_n}$ for $n \ge 1$. The sequence starts $1, \sqrt{3}, \sqrt{2+\sqrt{3}}, \dots$. We can show this sequence is increasing and bounded above by 2. Therefore, by the Monotone Convergence Theorem, it converges. Let the limit be $L$. Then $L = \sqrt{2+L}$, which implies $L^2 = 2+L$, or $L^2 - L - 2 = 0$. Factoring gives $(L-2)(L+1)=0$. Since the terms are positive, $L$ must be positive, so $L=2$. Thus, the sequence converges to 2.

Cauchy Sequences

While the definition of convergence is based on a limit $L$, there is an alternative way to characterize convergent sequences using the idea of terms getting close to each other, rather than close to a specific limit. This leads to the concept of a Cauchy sequence.

A sequence of real numbers $\{a_n\}$ is called a Cauchy sequence if for every positive number $\epsilon$, there exists a natural number $N$ such that for all $m, n > N$, the absolute difference $|a_m - a_n|$ is less than $\epsilon$.

In simpler terms, a Cauchy sequence is one where the terms eventually become arbitrarily close to each other.

The Cauchy Criterion for Convergence

The beauty of Cauchy sequences lies in the following theorem, known as the Cauchy criterion for convergence:

A sequence of real numbers converges if and only if it is a Cauchy sequence.

This is a powerful result because it allows us to determine if a sequence converges without needing to know or guess its limit beforehand. For the set of real numbers $\mathbb{R}$, this criterion is equivalent to the completeness property of $\mathbb{R}$.

Relationship Between Convergence and Cauchy Sequences

Let's understand why this equivalence holds:

1. If a sequence converges, then it is a Cauchy sequence.

Suppose $\lim_{n \to \infty} a_n = L$. Let $\epsilon > 0$ be given. Then there exists an $N$ such that for all $n > N$, $|a_n - L| < \frac{\epsilon}{2}$.

Now, let $m, n > N$. By the triangle inequality:

$|a_m - a_n| = |a_m - L + L - a_n| \le |a_m - L| + |L - a_n| = |a_m - L| + |a_n - L|$

Since $m > N$ and $n > N$, we have $|a_m - L| < \frac{\epsilon}{2}$ and $|a_n - L| < \frac{\epsilon}{2}$.

Therefore, $|a_m - a_n| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon$.

This shows that the sequence $\{a_n\}$ is a Cauchy sequence.

2. If a sequence is Cauchy, then it converges (for real numbers).

This part is more involved and relies on the completeness property of the real numbers. A constructive proof typically involves showing that a Cauchy sequence is bounded and then using the Monotone Convergence Theorem or Bolzano-Weierstrass Theorem to extract a convergent subsequence. The limit of this subsequence can then be shown to be the limit of the original Cauchy sequence.

For the purpose of exams, it's crucial to remember the definition of a Cauchy sequence and the theorem that states Cauchy sequences in $\mathbb{R}$ are precisely the convergent sequences.

Examples of Cauchy Sequences

Sequence: $a_n = \frac{1}{n}$. We showed this converges to 0. Therefore, it must be a Cauchy sequence. Let's verify: For $m, n > N$, $|\frac{1}{m} - \frac{1}{n}|$. Assume $m > n$. Then $|\frac{1}{m} - \frac{1}{n}| = \frac{1}{n} - \frac{1}{m} < \frac{1}{n}$. If we choose $N$ such that $\frac{1}{N} < \epsilon$, then for $n > N$, $\frac{1}{n} < \epsilon$. Since $m > n > N$, we have $|\frac{1}{m} - \frac{1}{n}| < \frac{1}{n} < \epsilon$.
Sequence: $a_n = \sum_{k=0}^{n} \frac{1}{k!}$. This sequence converges to $e$. Since it converges, it must be a Cauchy sequence.

Examples of Sequences That Are NOT Cauchy

Sequence: $a_n = n$. If we take $m, n$ large, say $m=1000, n=1001$, then $|a_m - a_n| = |1000 - 1001| = 1$. If we take $m=1000000, n=1000001$, then $|a_m - a_n| = 1$. The difference $|a_m - a_n|$ is always 1 for consecutive terms, so it never gets arbitrarily small. Thus, it's not a Cauchy sequence.
Sequence: $a_n = (-1)^n$. Consider $m$ even and $n$ odd. Then $|a_m - a_n| = |1 - (-1)| = 2$. This difference is always 2, not arbitrarily small. So, it's not a Cauchy sequence.

Key Takeaway: Convergence vs. Cauchy

A sequence converges if its terms approach a specific limit $L$. A sequence is Cauchy if its terms eventually get arbitrarily close to each other.

For Real Numbers ($\mathbb{R}$): These two concepts are equivalent. A sequence converges if and only if it is Cauchy.

For Other Number Systems (e.g., $\mathbb{Q}$): The equivalence does not always hold. For example, the sequence of rational numbers that approximates $\sqrt{2}$ is Cauchy but does not converge in $\mathbb{Q}$ because $\sqrt{2}$ is irrational.

Properties of Convergent Sequences

If $\{a_n\}$ converges to $L$ and $\{b_n\}$ converges to $M$, then:

$\{a_n + b_n\}$ converges to $L+M$.
$\{a_n - b_n\}$ converges to $L-M$.
$\{a_n b_n\}$ converges to $LM$.
$\{c a_n\}$ converges to $cL$ for any constant $c$.
If $M \ne 0$ and $b_n \ne 0$ for all $n$, then $\{\frac{a_n}{b_n}\}$ converges to $\frac{L}{M}$.

These properties are essential for evaluating limits of sequences. For example, to find the limit of $a_n = \frac{3n^2 + 1}{n^2 - 2}$, we can divide the numerator and denominator by the highest power of $n$ in the denominator ($n^2$):

$a_n = \frac{3 + \frac{1}{n^2}}{1 - \frac{2}{n^2}}$

We know $\lim_{n \to \infty} \frac{1}{n^2} = 0$ and $\lim_{n \to \infty} \frac{2}{n^2} = 0$. Using the limit properties:

$\lim_{n \to \infty} a_n = \frac{\lim_{n \to \infty} (3 + \frac{1}{n^2})}{\lim_{n \to \infty} (1 - \frac{2}{n^2})} = \frac{3 + 0}{1 - 0} = 3$.

The Squeeze Theorem (Sandwich Theorem) for Sequences

If we have three sequences $\{a_n\}$, $\{b_n\}$, and $\{c_n\}$ such that $a_n \le b_n \le c_n$ for all $n$ beyond some integer $N$, and if $\lim_{n \to \infty} a_n = L$ and $\lim_{n \to \infty} c_n = L$, then the sequence $\{b_n\}$ also converges to $L$.

Example: Find the limit of $b_n = \frac{\sin(n)}{n}$.

We know that $-1 \le \sin(n) \le 1$ for all $n$. Dividing by $n$ (assuming $n>0$):

$-\frac{1}{n} \le \frac{\sin(n)}{n} \le \frac{1}{n}$

Let $a_n = -\frac{1}{n}$ and $c_n = \frac{1}{n}$. We know that $\lim_{n \to \infty} a_n = 0$ and $\lim_{n \to \infty} c_n = 0$. By the Squeeze Theorem, $\lim_{n \to \infty} b_n = \lim_{n \to \infty} \frac{\sin(n)}{n} = 0$. This sequence is also Cauchy because it converges.

Summary Table

Concept	Definition	Key Property (for $\mathbb{R}$)	Example
Convergent Sequence	$\exists L$ such that for every $\epsilon > 0$, $\exists N$ where $n > N \implies \|a_n - L\| < \epsilon$.	Monotone and bounded sequences converge. Properties of arithmetic operations hold.	$a_n = 1/n \to 0$.
Divergent Sequence	A sequence that does not converge. Can diverge to $\pm \infty$ or oscillate.	Does not satisfy convergence criteria.	$a_n = n \to \infty$. $a_n = (-1)^n$ oscillates.
Cauchy Sequence	For every $\epsilon > 0$, $\exists N$ where $m, n > N \implies \|a_m - a_n\| < \epsilon$.	Equivalent to convergence for sequences in $\mathbb{R}$.	$a_n = \sum_{k=0}^{n} \frac{1}{k!}$ is Cauchy (converges to $e$).

Common Pitfalls and Important Notes

Confusing convergence with boundedness. A bounded sequence does not necessarily converge (e.g., $(-1)^n$).
Assuming a sequence converges just because its terms get smaller or larger. It must approach a *specific finite value* to converge.
Forgetting that the Cauchy criterion's equivalence to convergence is specific to complete metric spaces like $\mathbb{R}$.
Errors in applying limit properties, especially with indeterminate forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$ (which often require L'Hôpital's rule for functions, or more advanced sequence techniques).

Exam Tip: Proving Convergence without the Limit

If asked to prove a sequence converges, and you cannot easily guess the limit $L$, try to prove it is Cauchy. If you can show $|a_m - a_n|$ can be made arbitrarily small for large $m, n$, then you've proven convergence for real sequences!

Alternatively, if the sequence appears to be monotone and bounded, use the Monotone Convergence Theorem. This is often easier than direct epsilon-N proofs.