ISPA - Sequences

Sequences

Sequence Terms and Notation

A sequence is a list of numbers written in a definite order. Sequences can be either finite or infinite. An infinite sequence is of the general form equation image indicator . For every positive integer n, there is a corresponding number a_n. This sequence is defined as a function whose domain is the set of positive integers. Each of the numbers a₁, a₂, a_3, a₄, ... , a_n, a_n₊₁ ... are the terms of the sequence. The notation {a_n} or equation image indicator $LaTeX: \left\{a_n\right\}_{n=1}^{\infty}$ $\left\{a_n\right\}_{n=1}^{\infty}$ refers to a sequence whose n^th term is a_n. The integer n is called the index of a_n. It is possible to define some sequences such as $LaTeX: a_n=\left(\frac{\pi}{e}\right)^n=\left\{\frac{\pi}{e},\frac{\pi^2}{e^2},\frac{\pi^3}{e^3}...,\frac{\pi^n}{e^n},...\right\}$ $a_n=\left(\frac{\pi}{e}\right)^n=\left\{\frac{\pi}{e},\frac{\pi^2}{e^2},\frac{\pi^3}{e^3}...,\frac{\pi^n}{e^n},...\right\}$ by a formula for the n^th term. Other sequences such as the Fibonacci sequence {f_n} = {1, 1, 2, 3, 5, 8, 13, 21, ...}, the recursively defined sequence f₁ = 1, f₂ = 1, f_n = f_n-1 + f_n-2, n > 3, and {a_n} where a_n is the digit in the n^th decimal place of equation image indicator , {1, 4, 1, 5, 9, 2, 6, 5, 4, ...} are not definable by the nth term.

View the Introduction to Sequences presentation below.

Limit of a Sequence

The limit of a sequence {a_n} is defined as a real number L, written as equation image indicator $LaTeX: \lim_{n \to \infty }a_n=L\:\:or\:\:a_n\rightarrow L \:as\:n\rightarrow \infty$ $\lim_{n \to \infty }a_n=L\:\:or\:\:a_n\rightarrow L \:as\:n\rightarrow \infty$ , if the terms a_n can be made sufficiently close to L by taking n sufficiently large. If $LaTeX: \lim_{n \to \infty }a_n$ $\lim_{n \to \infty }a_n$ exists, the sequence converges to the number L. If no such limit exists, then the sequence diverges. It is also true that if equation image indicator $LaTeX: \lim_{n \to \infty }|a_n|=0$ $\lim_{n \to \infty }|a_n|=0$ , then $LaTeX: \lim_{n \to \infty }a_n=0$ $\lim_{n \to \infty }a_n=0$ .

Since sequences are functions with domain restricted to the positive integers, the limit laws for functions discussed in the Limits and Continuity module also hold for sequences.

image of different limit laws for sequences

The next presentation focuses on determining whether or not a sequence converges to a unique limiting value and illustrates how graphing a sequence on the TI-83/84Plus family can be used to support the existence or nonexistence of a unique limiting value.

L'Hopital's Rule

Because L'Hopital's Rule applies to functions and not to sequences, it is necessary to formalize the connection between equation image indicator $LaTeX: \lim_{a \to \infty }a_n$ $\lim_{a \to \infty }a_n$ and $LaTeX: \lim_{n \to \infty }f(x)$ $\lim_{n \to \infty }f(x)$ . Suppose that f(x) is a function defined for all x > n₀ and that {a_n} is a sequence of real numbers such that a_n = f(n) for n > n₀. Then equation image indicator $LaTeX: \lim_{n \to \infty }f(x)=L\Longrightarrow \lim_{a \to \infty }a_n=L$ $\lim_{n \to \infty }f(x)=L\Longrightarrow \lim_{a \to \infty }a_n=L$ . Consider the following example:

calculating the limit and L'Hopital's Rule example

Squeeze Theorem

In the Limits and Continuity module, the Squeeze Theorem for functions stated that if f(x) < g(x) < h(x) when x is near a (except for possibly at a) and equation image indicator $LaTeX: \lim_{x \to a}f(x)=\lim_{x \to a}h(x)=L\:then\:\lim_{x \to a}g(x)=L$ $\lim_{x \to a}f(x)=\lim_{x \to a}h(x)=L\:then\:\lim_{x \to a}g(x)=L$ . The Squeeze Theorem adapted for sequences states that if {a_n}, {b_n}, and {c_n} are sequences such that a_n < b_n < c_n for all n and equation image indicator $LaTeX: \lim_{n \to \infty }a_n=\lim_{n \to \infty }c_n=L, \:then\:\lim_{n \to \infty }b_n=L$ $\lim_{n \to \infty }a_n=\lim_{n \to \infty }c_n=L, \:then\:\lim_{n \to \infty }b_n=L$ .

View the next presentation on finding limits of a sequence using the Squeeze Theorem.

To explore the convergence and divergence of sequences using an interactive applet CLICK HERE. Links to an external site.

View the next presentation to learn how to graph sequences on a graphing calculator.

Monotonic and Bounded Sequences

A sequence {a_n} is increasing if a_n < a_n+1 for all n > 1 and it is decreasing if a_n > a_n+1 for all n > 1. A sequence that is either increasing or decreasing is a monotonic sequence. Consider examples of sequences that are either monotonic or not monotonic.

Example of monotonic versus not a monotonic sequence

A sequence {a_n} is bounded above if there is a number M such that a_n < M for all n > 1. The number M is called an upper bound of the sequence. If M is an upper bound for {a_n} but no number smaller than M is an upper bound for {a_n}, then M is a least upper bound for {a_n}.

A sequence {a_n} is bounded below if there is a number m such that m < a_n for all n > 1. The number m is called a lower bound of the sequence. If m is a lower bound for {a_n} but no number greater than m is a lower bound for {a_n}, then m is the greatest lower bound for {a_n}.

A sequence {a_n} that is bounded above and below is a bounded sequence. If {a_n} is not bounded, then {a_n} is an unbounded sequence. Combining the characteristics of monotonicity and boundedness, the Monotonic Sequence Theorem states that every bounded, monotonic sequence is convergent.

View the Monotonic and Bounded Sequences video and Slides 3 and 4 of the Sequences 2 video accessed from the More Resources sidebar section.

The next presentation shows how to prove a sequence is non-decreasing and bounded. Begin viewing at 4:10.

Sequences Practice

Sequences: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of sequences.

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