41. Sequences and Monotone Sequences

Dated: 30-10-2024

Definition of a Sequence

A sequence is a succession of numbers.

Example

\[1, 2, 3, \ldots\]

Each number is called term of the sequence.
Each term has a positional name as well such as 1st term, 2nd term etc.
We write them as

\[a_1, a_2, \ldots\]

Example

Let us say we have a sequence \(\(2, 4, 6, \ldots\)\)

Then we can also write it as

\[\{2^n\}^{+\infty}_{n = 1}\]

Generalized

Therefore, we can write a sequence of the form \(a_1, a_2, \ldots\) as

\[\{a_n\}^{+\infty}_{n = 1}\]

Formal Definition

A sequence or a infinite sequence is a function¹ whose domain is a set² of positive integers (\(\mathbb{Z^+}\)).
We have 2 notations for this

• \[\{a_n\}_{n = 1}^{+\infty}\]
• \[f(n) = a_n; n \in \mathbb{Z^+}\]

Graphs of Sequences

Imagine we have \(f(x) = x^2\)
Then if the domain¹ is \(x \in \{\mathbb{R^+} + \{0\}\}\) then the graph would look something like

But if the domain is \(x \in \{\mathbb{Z}^+ + \{0\}\}\) then graph would be

These graphs only are captured inside the interval³ \([0, 2]\)

Limits⁴ For Sequence

A sequence \(\{a_n\}\) is said to converge to the limit⁴ \(L\) if given any \(\epsilon > 0\) there is \(N\) such that \(\lvert a_n - L\rvert < \epsilon\) for \(n \ge N\).
In this case we write

\[\lim_{n \to + \infty} a_n = L\]

A sequence which does not converge is said to diverge.

Geometric Intuition

After \(n > 5\), our values start to approach \(L\).

Theorems

• \[\lim_{n \to +\infty} c = c\]
• \[\lim_{n \to +\infty}c a_n = c \lim_{n \to +\infty} a_n= c L_1\]
• \[\lim_{n \to +\infty} (a_n + b_n) = \lim_{n \to + \infty} a_n + \lim_{n \to +\infty} b_n = L_1 + L_2\]
• \[\lim_{n \to +\infty} (a_n - b_n) = \lim_{n \to + \infty} a_n - \lim_{n \to +\infty} b_n = L_1 - L_2\]
• \[\lim_{n \to +\infty} (a_n \cdot b_n) = \lim_{n \to + \infty} a_n \cdot \lim_{n \to +\infty} b_n = L_1 \cdot L_2\]
• \[\lim_{n \to +\infty}\left(\frac{a_n}{b_n}\right) = \frac{\lim_{n \to +\infty} a_n}{\lim_{n \to +\infty} b_n} = \frac{L_1}{L_2}\]

Example

Determine if the following sequence converges or diverges.

\[\left\{\frac{n}{2n+1}\right\}_{n=1}^{+\infty}\]

Solution

Divide the numerator and denominator by \(n\).

\[\lim_{n\rightarrow+\infty}\frac{n}{2n+1} = \lim_{n\rightarrow+\infty}\frac{1}{2+1/n} \]

\[= \frac{\lim_{n\rightarrow+\infty}1}{\lim_{n\rightarrow+\infty}(2+1/n)} = \frac{1}{2}\]

Recursive Sequence

There are some sequences which are defined by providing an initial value and by giving a formula which relates each subsequent term to the term that precedes it.
These sequences are called recursive sequences.

Example

For \(n \ge 1\)

\[a_{n+1} = \frac 1 2 \left(a_n + \frac {3}{a_n}\right)\]

Monotonicity

A sequence \(\{a_n\}\) is called

Increasing

If \(\(a_1 < a_2 < \ldots < a_n < \ldots\)\)

Difference of terms

\[a_{n + 1} - a_n > 0\]

Ratio of terms

\[\frac{a_{n + 1}}{a_n} > 1\]

Derivative of \(f(x) = a_n\)

\[f^{\prime}(x) > 0\]

Decreasing

If \(\(a_1 > a_2 > \ldots > a_n > \ldots\)\)

Difference of terms

\[a_{n + 1} - a_n < 0\]

Ratio of terms

\[\frac{a_{n + 1}}{a_n} < 1\]

Derivative of \(f(x) = a_n\)

\[f^{\prime}(x) < 0\]

Non Increasing

If \(\(a_1 \ge a_2 \ge \ldots \ge a_n \ge \ldots\)\)

Difference of terms

\[a_{n + 1} - a_n \le 0\]

Ratio of terms

\[\frac{a_{n + 1}}{a_n} \le 1\]

Derivative of \(f(x) = a_n\)

\[f^{\prime}(x) \le 0\]

Non Decreasing

If \(\(a_1 \le a_2 \le \ldots \le a_n \le \ldots\)\)

Difference of terms

\[a_{n + 1} - a_n \ge 0\]

Ratio of terms

\[\frac{a_{n + 1}}{a_n} \ge 1\]

Derivative of \(f(x) = a_n\)

\[f^{\prime}(x) \ge 0\]

A sequence that is either non decreasing or non increasing is called a monotone.
A sequence that is either decreasing or increasing is called a strictly monotone.

Example

Prove that the sequence

\[\frac 1 2, \frac 2 3, \ldots , \frac{n}{n+1}, \ldots\]

is an increasing sequence.

Solution

\[a_n = \frac{n}{n+1}\]

\[a_{n + 1} = \frac{n + 1}{n+2}\]

Now from here, we can take either the difference route or ratio route.

Difference

\[a_{n+1} - a_n = \frac{n+1}{n+2} - \frac{n}{n+1}\]

\[= \frac{n^2 + 2n + 1 - n^2 - 2n}{(n+1)(n+2)}\]

\[= \frac{1}{(n+1)(n+2)} > 0\]

Ratio

\[\frac{a_{n+1}}{a_n} = \frac{(n+1)/(n+2)}{n/(n+1)}\]

\[= \frac{n+1}{n+2} \cdot \frac{n+1}{n} \]

\[= \frac{n^2 + 2n + 1}{n^2 + 2n} > 1\]

Eventually Monotonic

A sequence is called eventually monotonic for some \(n = N\), the sequence becomes a monotone.

Theorems

Non Decreasing

1. There is a constant \(M\) such that \(a_n \le M\) for all \(n\) then the sequence converges to a limit⁴ satisfying \(L \le M\).

2. There is no upper bound in case of

\[\lim_{n \to +\infty}a_n = +\infty\]

Non Increasing

1. There is a constant \(M\) such that \(a_n \ge M\) for all \(n\) then the sequence converges to a limit⁴ satisfying \(L \ge M\).

2. There is no lower bound in case of

\[\lim_{n \to +\infty}a_n = -\infty\]

References

Read more about notations and symbols.

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1. Read more about functions. ↩↩

2. Read more about sets. ↩

3. Read more about interval. ↩

4. Read more about limits. ↩↩↩↩