18. Sequences

Sometimes we are unable to compute a limit exactly. However, if we can at least show that the limit exists, then we can use some numerical technique to approximate the limit. We now introduce two properties of sequences (bounded and monotonic) which together help us show the limit exists.

c1. Bounded Sequences

Bounded Sequences

A sequence ana_n is bounded above if there is a number MM such that anMa_n \le M for all nn. Then MM is called an upper bound.

A sequence ana_n is bounded below if there is a number mm such that anma_n \ge m for all nn. Then mm is called an lower bound.

A sequence ana_n is bounded if it is both bounded above and bounded below.

Here are some examples:

bnd1
an=3(1+(1)n)na_n=3-(1+(-1)^n)n
Bounded above by 33.
Unbounded below.
bnd3
an=sinna_n=\sin n
Bounded above by 11.
Bounded below by 1-1.
bnd2
an=3+(1+(1)n)na_n=3+(1+(-1)^n)n
Bounded below by 33.
Unbounded above.
bnd4
an=(1)nna_n=(-1)^n n
Unbounded above.
Unbounded below.

Determine if each sequence is bounded above, below, neither or both. When appropriate, find upper and lower bounds.

  1. an=21na_n=2-\dfrac{1}{n}

    Answer

    Bounded below by 11 and bounded above by 22.

    [×]

    Solution

    Since n1,n \ge 1, then 0<1n10<\dfrac{1}{n}\le1 and 0>1n10>-\dfrac{1}{n}\ge-1 and 2>21n12>2-\dfrac{1}{n}\ge1.
    So {21n}\left\{ 2-\dfrac{1}{n}\right\} is bounded below by 11 and bounded above by 22.

    [×]
  2. an=(1)nnna_n=(-1)^n n-n

    Answer

    Bounded above by 00 but not bounded below.

    [×]

    Solution

    We study the sequence an=(1)nnna_n=(-1)^n n-n separately for nn odd and nn even.

    For nn odd, (1)n=1(-1)^n=-1 and an=2na_n=-2n which is always negative and becomes arbitrarily large and negative.

    For nn even, (1)n=1(-1)^n=1 and an=nn=0a_n=n-n=0.

    So the sequence is bounded above by 00 (or by any positive number) but the sequence is not bounded below and so is not bounded.

    [×]
  3. an=n!2na_n=\dfrac{n!}{2^n}

    Answer

    Bounded below by 00 (or 12\dfrac{1}{2}) but unbounded above.

    [×]

    Solution

    Since the terms an=n!2na_n=\dfrac{n!}{2^n} are all positive, the sequence is bounded below by 00.

    To see if there is an upper bound, we write the nthn^\text{th} term as: an=12223242n12n2=an1n2 a_n=\dfrac{1}{2}\cdot\dfrac{2}{2}\cdot\dfrac{3}{2}\cdot\dfrac{4}{2} \cdot\,\cdots\,\cdot\dfrac{n-1}{2}\cdot\dfrac{n}{2} =a_{n-1}\cdot\dfrac{n}{2} Thus as long as n>2n \gt 2, the term ana_n is the term an1a_{n-1} times something bigger than 11, namely n2\dfrac{n}{2}. So we suspect the sequence is unbounded above.

    To see this we use a proof by contradiction. Suppose there is an upper bound MM. Then the term a4Ma_{4M} is a4M=122232424M124M2=M2232424M12\begin{aligned} a_{4M} &=\dfrac{1}{2}\cdot\dfrac{2}{2}\cdot\dfrac{3}{2}\cdot \dfrac{4}{2}\cdot\,\cdots\,\cdot \dfrac{4M-1}{2}\cdot\dfrac{4M}{2} \\ &=M\cdot\dfrac{2}{2}\cdot\dfrac{3}{2}\cdot \dfrac{4}{2}\cdot\,\cdots\,\cdot \dfrac{4M-1}{2} \end{aligned} which is MM times a number bigger than 11. So a4M>Ma_{4M} \gt M which contradicts the assumption that MM is an upper bound.

    [×]

© 2025 MYMathApps

Supported in part by NSF Grant #1123255