Limits of Sequences

1Introduction

This lecture defines convergence of a sequence to a real number by the $\epsilon$\varepsilon-$N$N criterion and develops the following principle: a limit is determined not by finitely many initial terms, but by all sufficiently late terms at every requested accuracy.

To analyze the long-term behavior of solutions to difference equations and the partial sums used in the next lecture on series, one must quantify what it means for a sequence to approach a value. We consider real sequences $(a_n{)}_{n\ge 1}$(a_n)_{n\geq1} and assume the standard properties of absolute value.

2Terms and Definitions

A sequence $\left(a_n\right)$(a_n) {converges} to $L\in \mathbb{R}$L\in\mathbb R if

We then write $\underset{n\to \mathrm{\infty }}{\lim }{a}_n=L$\lim_{n\to\infty}a_n=L or $a_n\to L$a_n\to L. The number $\epsilon$\varepsilon is the requested error tolerance. The index $N$N may depend on that tolerance, but after $N$N has been chosen, the same estimate must hold for every $n\ge N$n\geq N.

A sequence {diverges} if it does not converge to any finite real number. In particular, we write $a_n\to +\mathrm{\infty }$a_n\to+\infty if

Similarly, $a_n\to -\mathrm{\infty }$a_n\to-\infty means that all sufficiently late terms lie below every prescribed real bound $M$M. The symbols $+\mathrm{\infty }$+\infty and $-\mathrm{\infty }$-\infty are not real limit values; they denote particular modes of divergence. For example, $(-1{)}^n$(-1)^n diverges by oscillation and tends to neither $+\mathrm{\infty }$+\infty nor $-\mathrm{\infty }$-\infty.

3Basic Properties

A convergent sequence has a unique limit. Indeed, suppose $a_n\to L$a_n\to L and $a_n\to K$a_n\to K with $L\ne K$L\ne K. Apply both definitions with $\epsilon =|L-K|/3$\varepsilon=|L-K|/3 and choose one sufficiently large $n$n. The triangle inequality gives

a contradiction.

Every convergent sequence is also bounded. If $a_n\to L$a_n\to L, choosing $\epsilon =1$\varepsilon=1 shows that all sufficiently late terms satisfy $\left|a_n\right|\le \left|L\right|+1$|a_n|\leq|L|+1. The finitely many remaining terms have a maximum absolute value, so one constant bounds the entire sequence. Consequently, every unbounded sequence diverges. Boundedness alone does not guarantee convergence, as $(-1{)}^n$(-1)^n demonstrates.

We shall use the standard limit laws. If $a_n\to A$a_n\to A and $b_n\to B$b_n\to B, then

In the quotient law, $b_n\ne 0$b_n\ne0 for all sufficiently large $n$n. These laws follow from the $\epsilon$\varepsilon-$N$N definition by splitting errors with the triangle inequality and using boundedness of convergent sequences.

4Geometric Sequences

Let $a_n=a{r}^{n-1}$a_n=ar^{n-1}. If $a=0$a=0, the sequence is identically zero. Assume below that $a\ne 0$a\ne0.

• If $\left|r\right|<1$|r|<1, then $a_n\to 0$a_n\to0. For $0<\left|r\right|<1$0<|r|<1, put $q=1/\left|r\right|>1$q=1/|r|>1. The binomial theorem gives $q^m=(1+(q-1){)}^m\ge 1+m(q-1)$q^m=(1+(q-1))^m\geq1+m(q-1). Given $\epsilon >0$\varepsilon>0, choose $m$m so that $1+m(q-1)>\left|a\right|/\epsilon$1+m(q-1)>|a|/\varepsilon. Set $N=m+1$N=m+1; if $n\ge N$n\geq N, then $\left|a{r}^{n-1}\right|\le \left|a{r}^m\right|<\epsilon$|ar^{n-1}|\leq|ar^m|<\varepsilon. The case $r=0$r=0 is immediate.
• If $r=1$r=1, then $a_n=a$a_n=a, so the sequence converges to $a$a.
• If $r=-1$r=-1, the terms alternate between $a$a and $-a$-a. If $a_n\to L$a_n\to L, applying the definition along the even and odd indices would make both constant subsequences $a$a and $-a$-a converge to $L$L, contrary to uniqueness and $a\ne 0$a\ne0. Thus the sequence diverges.
• If $r>1$r>1, the same binomial estimate shows that $r^{n-1}$r^{n-1} eventually exceeds every positive number. Hence $a_n\to +\mathrm{\infty }$a_n\to+\infty when $a>0$a>0, and $a_n\to -\mathrm{\infty }$a_n\to-\infty when $a<0$a<0.
• If $r\le -1$r\leq-1, the sign alternates. When $r<-1$r<-1, the absolute values are also unbounded, so the sequence diverges and tends to neither $+\mathrm{\infty }$+\infty nor $-\mathrm{\infty }$-\infty.

Thus, for nonzero real $a$a, a finite limit exists exactly when $\left|r\right|<1$|r|<1 or $r=1$r=1.

5Sequences Defined by Rational Expressions

Let $P$P and $Q$Q be nonzero real polynomials, assume $Q\left(n\right)\ne 0$Q(n)\ne0 for all sufficiently large $n$n, and set

Write $p=\text{[PARSE ERROR: Undefined("Command("deg")")]}P$p=\deg P and $q=\text{[PARSE ERROR: Undefined("Command("deg")")]}Q$q=\deg Q, and let their leading coefficients be $c$c and $d$d, respectively. For $p\le q$p\leq q, divide the numerator and denominator by $n^q$n^q. For $p>q$p>q, write $a_n/n^{p-q}=P\left(n\right)/\left(n^{p-q}Q\right(n\left)\right)$a_n/n^{p-q}=P(n)/(n^{p-q}Q(n)) and divide its numerator and denominator by $n^p$n^p. Applying the limit laws together with $1/n^k\to 0$1/n^k\to0 yields:

• If $p<q$p<q, then $a_n\to 0$a_n\to0.
• If $p=q$p=q, then $a_n\to c/d$a_n\to c/d.
• If $p>q$p>q, then $a_n/n^{p-q}\to c/d$a_n/n^{p-q}\to c/d. Sufficiently late values of this ratio have the same sign as $c/d$c/d and absolute value at least $|c/(2d\left)\right|$|c/(2d)|, while $n^{p-q}$n^{p-q} eventually exceeds every positive number. Hence $a_n\to +\mathrm{\infty }$a_n\to+\infty when $c/d>0$c/d>0 and $a_n\to -\mathrm{\infty }$a_n\to-\infty when $c/d<0$c/d<0.

For example,

The denominator tends to $1\ne 0$1\ne0, so the quotient law applies.

6Squeeze Theorem

Suppose that for every sufficiently large $n$n,

and that $b_n\to L$b_n\to L and $c_n\to L$c_n\to L. Then $a_n\to L$a_n\to L. Indeed, for every $\epsilon >0$\varepsilon>0, sufficiently large $n$n satisfy $L-\epsilon <b_n$L-\varepsilon<b_n and $c_n<L+\epsilon$c_n<L+\varepsilon, and hence $|a_n-L|<\epsilon$|a_n-L|<\varepsilon.

Since $\left|\sin n\right|\le 1$|\sin n|\leq1 and $1/n\to 0$1/n\to0,

The squeeze theorem therefore gives $\sin n/n\to 0$\sin n/n\to0. No convergence assumption on $\sin n$\sin n itself is required.

7Decision Procedure

• First distinguish among a candidate finite limit, divergence to infinity, and oscillation.
• For a geometric factor, separate the cases $\left|r\right|<1$|r|<1, $r=1$r=1, $r=-1$r=-1, and $\left|r\right|>1$|r|>1.
• For a rational expression, compare the degrees and leading coefficients of its numerator and denominator.
• When an absolute value can be bounded by a convergent upper bound, apply the squeeze theorem.
• When invoking a limit law, verify its assumptions, especially the nonvanishing denominator in the quotient law.

8Scope

Because the index in the $\epsilon$\varepsilon-$N$N definition ranges only over natural numbers, there is no input-distance parameter $\delta$\delta such as the one used in the $\epsilon$\varepsilon-$\delta$\delta definition of a function limit. Although a sequence may be regarded as the values $f\left(n\right)$f(n) of a function at integer points, the behavior of $f$f between those points does not affect convergence of the sequence.

This lecture concerns real sequences. Convergence of complex sequences is defined analogously by replacing absolute value with the complex modulus, but the ordered classification by $+\mathrm{\infty }$+\infty and $-\mathrm{\infty }$-\infty does not transfer unchanged.

9Basic Formulas

10Summary

• A sequence converges when all sufficiently late terms meet every requested error tolerance.
• Divergence includes divergence to infinity and oscillation; $\pm \mathrm{\infty }$\pm\infty are not finite limit values.
• Geometric sequences, rational expressions, and squeezing reduce long-term behavior to absolute values, leading terms, and inequalities, respectively.

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