For the more pedantic reader, here we mean a countably infinite list of numbers. The interested (pedantic or otherwise) reader should look up countable and uncountable sets.
many numbers with a specified order. It is denoted
It is not necessary that there be a simple explicit formula for the \(n^{\rm th}\) term of a sequence. For example the decimal digits of \(\pi\) is a perfectly good sequence
There is, however, a remarkable result due to Bailey, Borwein and Plouffe that can be used to compute the \(n^{\rm th}\) binary digit of \(\pi\) (i.e. writing \(\pi\) in base 2 rather than base 10) without having to work out the preceding digits.
Our primary concern with sequences will be the behaviour of \(a_n\) as \(n\) tends to infinity and, in particular, whether or not \(a_n\) “settles down” to some value as \(n\) tends to infinity.
A sequence \(\big\{a_n\big\}_{n=1}^\infty\) is said to converge to the limit \(A\) if \(a_n\) approaches \(A\) as \(n\) tends to infinity. If so, we write
\begin{equation*}
\lim_{n\rightarrow\infty} a_n=A\qquad\hbox{or}\qquad a_n\rightarrow A\text{ as }n\rightarrow\infty
\end{equation*}
A sequence is said to converge if it converges to some limit. Otherwise it is said to diverge.
Three of the four sequences in Example 3.1.2 diverge:
The sequence \(\big\{a_n=n\big\}_{n=1}^\infty\) diverges because \(a_n\) grows without bound, rather than approaching some finite value, as \(n\) tends to infinity.
The sequence \(\big\{a_n=(-1)^{n-1}\big\}_{n=1}^\infty\) diverges because \(a_n\) oscillates between \(+1\) and \(-1\) rather than approaching a single value as \(n\) tends to infinity.
The sequence of the decimal digits of \(\pi\) also diverges, though the proof that this is the case is a bit beyond us right now 3
If the digits of \(\pi\) were to converge, then \(\pi\) would have to be a rational number. The irrationality of \(\pi\) (that it cannot be written as a fraction) was first proved by Lambert in 1761. Niven’s 1947 proof is more accessible and we invite the interested reader to use their favourite search engine to find step-by-step guides to that proof.
As \(n\rightarrow\infty\text{,}\) the \(\frac{1}{n}\) in the denominator tends to zero, so that the denominator \(2+\frac{1}{n}\) tends to \(2\) and \(\frac{1}{2+\frac{1}{n}}\) tends to \(\frac{1}{2}\text{.}\) So
Notice that in this last example, we are really using techniques that we used before to study infinite limits like \(\ds \lim_{x\rightarrow\infty}f(x)\text{.}\) This experience can be easily transferred to dealing with \(\lim\limits_{n\rightarrow\infty}a_n\) limits by using the following result.
The bulk of the rules for the arithmetic of limits of functions that you already know also apply to the limits of sequences. That is, the rules you learned to work with limits such as \(\ds \lim_{x\rightarrow\infty}f(x)\) also apply to limits like \(\ds\lim_{n\rightarrow\infty}a_n\text{.}\)
Let \(A\text{,}\)\(B\) and \(C\) be real numbers and let the two sequences \(\big\{a_n\big\}_{n=1}^\infty\) and \(\big\{b_n\big\}_{n=1}^\infty\) converge to \(A\) and \(B\) respectively. That is, assume that
We use these rules to evaluate limits of more complicated sequences in terms of the limits of simpler sequences — just as we did for limits of functions.
Let \(\{a_n\}_{n=1}^{\infty}\text{,}\)\(\{b_n\}_{n=1}^{\infty}\text{,}\) and \(\{c_n\}_{n=1}^{\infty}\text{,}\) be sequences with \(\lim\limits_{n \to \infty}a_n=A\text{,}\)\(\lim\limits_{n \to \infty}b_n=B\text{,}\) and \(\lim\limits_{n \to \infty}c_n=C\text{.}\) Assume \(A\text{,}\)\(B\text{,}\) and \(C\) are nonzero real numbers.
The limits of the sequences below can be evaluated using the squeeze theorem. For each sequence, choose an upper bounding sequence and lower bounding sequence that will work with the squeeze theorem.
Below is a list of sequences, and a list of functions.
Match each sequence \(\big\{a_n\big\}_{n=1}^\infty\) to any and all functions \(f(x)\) such that \(f(n)=a_n\) for all positive whole numbers \(n\text{.}\)
Match each sequence \(\big\{a_n\big\}_{n=1}^\infty\) to any and all functions \(f(x)\) such that \(\displaystyle\lim_{n \to \infty}a_n = \lim_{x \to \infty}f(x)\text{.}\)
Let \(\{a_n\}_{n=1}^\infty\) be a sequence defined by \(a_n = \cos n\text{.}\)
Give three different whole numbers \(n\) that are within 0.1 of an odd integer multiple of \(\pi\text{,}\) and find the corresponding values of \(a_n\text{.}\)
Remark: this demonstrates intuitively, though not rigorously, why \(\lim\limits_{n \to \infty}\cos n\) is undefined. We consistently find terms in the series that are close to \(-1\text{,}\) and also consistently find terms in the series that are close to 0. Contrast this to a series like \(\big\{\cos(2\pi n)\big\}\text{,}\) whose terms are always 1, and whose limit therefore is 1. It is possible to turn the ideas of this question into a rigorous proof that \(\lim\limits_{n \to \infty}\cos n\) is undefined. See the solution.
Consider the sequence \(\Big\{(-1)^n\sin\big(\frac{1}{n}\big)\Big\}\text{.}\) State whether this sequence converges or diverges, and if it converges give its limit.
Let \(\{A_n\}_{n=3}^\infty\) be the area of a regular polygon with \(n\) sides, with the distance from the centroid of the polygon to each corner equal to 1.
For a fixed constant \(x \ge 1\text{,}\)\(\{f_n\}\) is the sequence \(\{0,0,0,\ldots,0,1,0,\ldots,0,0,0,\ldots\}\text{.}\) The sole nonzero element comes in position \(k\text{,}\) where \(k\) is what we get when we round \(x\)down to a whole number. If \(x \lt 1\text{,}\) then the sequence consists of all zeroes.
Since we can plug in different values of \(x\text{,}\) we can think of \(f_n(x)\) as a function of sequences: a different \(x\) gives you a different sequence. On the other hand, if we imagine fixing \(n\text{,}\) then \(f_n(x)\) is just a function, where \(f_n(x)\) gives the \(n\)th term in the sequence corresponding to \(x\text{.}\)
Suppose the sequence \(\{w_1,w_2,w_3,\ldots\}\) is a list of all words in a language, where \(w_n\) is the word that is the \(n\)th most frequently used. Let \(f_n\) be the frequency of word \(w_n\text{.}\) Is \(\{f_1,f_2,f_3,\ldots\}\) an increasing sequence or a decreasing sequence?
Suppose in a language, \(w_1\) (the most frequently used word) has frequency \(6\%\text{.}\) If the language follows Zipf’s Law, then what frequency does \(w_3\) have?
The word “the” is the most-used word in contemporary American English. In a collection of about 450 million words, “the” appeared 22,038,615 times. The second-most used word is “be,” followed by “and.” About how many usages of these words do you expect in the same collection of 450 million words?
