2.3: Monotone Sequences - Mathematics LibreTexts

an≤an+1 for all n∈N. ... an≥an+1 for all n∈N. If {an} is increasing or decreasing, then it is called a monotone sequence. The sequence is ... Skiptomaincontent Definition\(\PageIndex{1}\)Theorem\(\PageIndex{1}\)-MonotoneConvergenceTheorem.Remark\(\PageIndex{2}\)Example\(\PageIndex{1}\)Example\(\PageIndex{2}\)Example\(\PageIndex{3}\)-Thenumbere.Theorem\(\PageIndex{3}\)-NestedIntervalsTheorem.Definition\(\PageIndex{2}\)Remark\(\PageIndex{4}\)Example\(\PageIndex{4}\)Theorem\(\PageIndex{5}\)Theorem\(\PageIndex{6}\)Theorem\(\PageIndex{7}\) Definition\(\PageIndex{1}\) Asequence\(\left\{a_{n}\right\}\)iscalledincreasingif \[a_{n}\leqa_{n+1}\text{forall}n\in\mathbb{N}.\] Itiscalleddecreasingif \[a_{n}\geqa_{n+1}\text{forall}n\in\mathbb{N}.\] If\(\left\{a_{n}\right\}\)isincreasingordecreasing,thenitiscalledamonotonesequence. Thesequenceiscalledstrictlyincreasing(resp.strictlydecreasing)if\(a_{n}a_{n+1}\text{forall}n\in\mathbb{N}\). Itiseasytoshowbyinductionthatif\(\left\{a_{n}\right\}\)isanincreasingsequence,then\(a_{n}\leqa_{m}\)whenever\(n\leqm\). Theorem\(\PageIndex{1}\)-MonotoneConvergenceTheorem. Let\(\left\{a_{n}\right\}\)beasequenceofrealnumbers.Thefollowinghold: If\(\left\{a_{n}\right\}\)isincreasingandboundedabove,thenitisconvergent. If\(\left\{a_{n}\right\}\)isdecreasingandboundedbelow,thenitisconvergent. Proof (a)Let\(\left\{a_{n}\right\}\)beanincreasingsequencethatisboundedabove.Define \(A=\left\{a_{n}:n\in\mathbb{N}\right\}\). Then\(A\)isasubsetof\(\mathbb{R}\)thatisnonemptyandboundedaboveand,hence,\(\supA\)exists.Let\(\ell=\supA\)andlet\(\varepsilon>0\).ByProposition1.5.1,thereexists\(N\in\mathbb{N}\)suchthat \(\ell-\varepsilon2=a_{1}\),thestatementistruefor\(n=1\).Next,suppose\(a_{k}M\text{forall}n\geqN.\] Inthiscase,wewrite\(\lim_{n\rightarrow\infty}a_{n}=\infty\).Similarly,wesaythat\(\left\{a_{n}\right\}\)divergesto\(-\infty\)andwrite\(\lim_{n\rightarrow\infty}a_{n}=-\infty\)ifforevery\(M\in\mathbb{R}\),thereexists\(N\in\mathbb{N}\)suchthat \[a_{n}5M\).Then,if\(n\geqN\),wehave \[a_{n}\geq\frac{n}{5}\geq\frac{N}{5}>M.\] Thefollowingresultcompletesthedescriptionofthebehaviorofmonotonesequences. Theorem\(\PageIndex{5}\) Ifasequence\(\left\{a_{n}\right\}\)isincreasingandnotboundedabove,then \[\lim_{n\rightarrow\infty}a_{n}=\infty.\] Similarly,if\(\left\{a_{n}\right\}\)isdecreasingandnotboundedbelow,then \[\lim_{n\rightarrow\infty}a_{n}=-\infty\] Proof Fixanyrealnumber\(M\).Since\(\left\{a_{n}\right\}\)isnotboundedabove,thereexists\(N\in\mathbb{N}\)suchthat\(a_{N}\geqM\).Then \[a_{n}\geqa_{N}\geqM\text{forall}n\geqN\] because\(\left\{a_{n}\right\}\)isincreasing.Therefore,\(\lim_{n\rightarrow\infty}a_{n}=\infty\).Theproofforthesecondcaseissimilar.\(\square\) Theorem\(\PageIndex{6}\) Let\(\left\{a_{n}\right\}\),\(\left\{b_{n}\right\}\),and\(\left\{c_{n}\right\}\)besequencesofrealnumbersandlet\(k\)beaconstant.Suppose \[\lim_{n\rightarrow\infty}a_{n}=\infty,\lim_{n\rightarrow\infty}b_{n}=\infty,\text{and}\lim_{n\rightarrow\infty}c_{n}=-\infty\] Then \(\lim_{n\rightarrow\infty}\left(a_{n}+b_{n}\right)=\infty\); \(\lim_{n\rightarrow\infty}\left(a_{n}b_{n}\right)=\infty\); \(\lim_{n\rightarrow\infty}\left(a_{n}c_{n}\right)=-\infty\); \(\lim_{n\rightarrow\infty}ka_{n}=\infty\)if\(k>0\),and\(\lim_{n\rightarrow\infty}ka_{n}=-\infty\)if\(k<0\); \(\lim_{n\rightarrow\infty}\frac{1}{a_{n}}=0\).(Hereweassume\(a_{n}\neq0\)forall\(n\).) Proof Weprovideproofsfor(a)and(e)andleavetheothersasexercises. (a)Fixany\(M\in\mathbb{R}\).Since\(\lim_{n\rightarrow\infty}a_{n}=\infty\),thereexists\(N_{1}\in\mathbb{N}\)suchthat \[a_{n}\geq\frac{M}{2}\text{forall}n\geqN_{1}.\] Similarly,thereexists\(N_{2}\in\mathbb{N}\)suchthat \[b_{n}\geq\frac{M}{2}\text{forall}n\geqN_{1}\] Let\(N=\max\left\{N_{1},N_{2}\right\}\).Thenitisclearthat \[a_{n}+b_{n}\geqM\text{forall}n\geqN.\] Thisimplies(a). (e)Forany\(\varepsilon>0\),let\(M=\frac{1}{\varepsilon}\).Since\(\lim_{n\rightarrow\infty}a_{n}=\infty\),thereexists\(N\in\mathbb{N}\)suchthat \[a_{n}>\frac{1}{\varepsilon}\text{forall}n\geqN\] Thisimpliesthatfor\(n\geqN\), \[\left|\frac{1}{a_{n}}-0\right|=\frac{1}{a_{n}}0\). \(a_{n+1}=\frac{1}{2}\left(a_{n}+\frac{b}{a_{n}}\right),b>0\). Answer Addtextshere.Donotdeletethistextfirst. Exercise\(\PageIndex{3}\) Provethateachofthefollowingsequencesisconvergentandfinditslimit. \(\sqrt{2};\sqrt{2\sqrt{2}};\sqrt{2\sqrt{2\sqrt{2}}};\cdots\) \(1/2;\frac{1}{2+1/2};\frac{1}{2+\frac{1}{2+1/2}};\cdots\) Answer Addtextshere.Donotdeletethistextfirst. Exercise\(\PageIndex{4}\) Provethatthefollowingsequenceisconvergent: \(a_{n}=1+\frac{1}{2!}+\frac{1}{3!}+\cdots+\frac{1}{n!},n\in\mathbb{N}\). Answer Addtextshere.Donotdeletethistextfirst. Exercise\(\PageIndex{5}\) Let\(a\)and\(b\)betwopositiverealnumberswith\(a