Math 2565, Spring 2020

Sequences

February 27, 2020

Sean Fitzpatrick
University of Lethbridge

Warm-Up

  • Write out the first 5 terms in each sequence:

    1. \(\di a_n = \frac{2^n}{n}\)

    2. \(\di b_n = \frac{(-1)^n n^2}{n!}\)

    3. \(\di c_n = n\cos(\pi n)\)

  • Determine a formula for the sequence:

    1. \(2, 4, 8, 16, 32, \ldots\)

    2. \(\di 1,-\frac23,\frac45,-\frac67,\frac89,\ldots\)

    3. \(\di \frac{1}{3}, \frac{3}{8}, \frac{1}{3}, \frac{7}{24}, \frac{9}{35},\ldots\)

Quiz

What does it mean to say that a sequence \(\{a_n\}\text{:}\)

  1. converges?

  2. is monotonic?

Sequences

  • A sequence is a function \(f\) from \(\mathbb{N}\) to \(\mathbb{R}\text{.}\)

  • We usually write \(a_n=f(n)\) for a single term, and either \(\{a_n\}\) or \((a_n)\) for the whole sequence.

  • Can think of a sequence as an “infinite list”: \((a_1,a_2,a_3,a_4,\ldots)\)

  • Sequences can be defined by formula: e.g. \(a_n = n^2/2^n\)

  • or recursively: e.g. \(a_1=1, a_2=1\text{,}\) and, for \(n\geq 1\text{,}\) \(a_{n+2}=a_n+a_{n+1}\text{.}\)

Examples

  1. Arithmetic: \(a_1=b\text{,}\) \(a_{n+1}=a_n+d\)

  2. Geometric: \(a_1 = k\text{,}\) \(a_{n+1}=ra_n\)

  3. Factorial: \(a_0=1\text{,}\) \(a_{n}=na_{n-1}\)

  4. And so many more

Limits

Definition:

Let \(\{a_n\}\) be a sequence. We say that the sequence converges to a limit \(L\text{,}\) and write

\begin{equation*} \lim_{n\to\infty}a_n = L\text{,} \end{equation*}
if for all \(\varepsilon>0\text{,}\) there exists an integer \(N>0\text{,}\) such that, for all \(n\geq N\text{,}\) \(\lvert a_n-L\rvert \lt \varepsilon\text{.}\)

If \(L=\infty\) or does not exist, we say the sequence diverges.

Note: if \(a_n=f(n)\) for some function \(f\) defined on \(\mathbb{R}\text{,}\) then

\begin{equation*} \lim_{n\to \infty}a_n = \lim_{x\to \infty}f(x)\text{.} \end{equation*}

Examples

Find the limit of the sequence:

  1. \(\di a_n = \frac1n\) (Assumes the Archimedian property of \(\R\text{.}\))

  2. \(\di a_n = r^n\text{,}\) if:

    • \(r\gt 1\)

    • \(r=1\)

    • \(0\lt r \lt 1\)

  3. \(\di a_n = \frac{2+4n-5n^2}{2n^2+9n}\)

More examples

Keep finding those limits:

  1. \(a_n = \cos(\pi n/3)\)

  2. \(a\di _n = \frac{n}{n+1}-\frac{n+1}{n}\)

  3. \(\di a_n = \left(1+\frac{3}{n}\right)^n\)

  4. \(\di a_n = \frac{2n}{\sqrt{1+9n^2}}\)

Properties

Limit properties are pretty much what you expect. Assume \(\{a_n\}\) and \(\{b_n\}\) converge. Then:

  • \(\di \lim_{n\to\infty}(a_n\pm b_n) = \lim_{n\to \infty}a_n \pm \lim_{n\to\infty}b_n\)

  • \(\di \lim_{n\to \infty}ca_n = c\lim_{n\to \infty}a_n\)

  • \(\di \lim_{n\to \infty}(a_nb_n) = \lim_{n\to \infty}a_n\cdot\lim_{n\to\infty}b_n\)

  • If \(f\) is continuous, \(\lim\limits_{n\to \infty}f(a_n)=f\left(\lim\limits_{n\to\infty}a_n\right)\text{.}\)

Bounded and monotone sequences

Definition:

A sequence \(\{a_n\}\) is bounded if \(m\lt a_n \lt M\) for some real \(m,M\text{.}\)

A sequence is monotone increasing if \(a_n\lt a_{n+1}\) for all \(n\text{.}\)

A sequence is monotone decreasing if \(a_n\gt a_{n+1}\) for all \(n\text{.}\)

Theorem:

  1. Any convergent sequence is bounded.

  2. Any bounded, monotone sequence converges.

Examples

Of the following sequences, which are bounded? Which are montone?

  1. \(\di a_n = \frac{5}{2n+1}\)

  2. \(\di a_n = \frac{2n-1}{3n+5}\)

  3. \(\di a_n = \sin(n\pi/4)\)

  4. \(\di a_n = \frac{n^2-4}{3n}\)

Example

Let \(\{a_n\}\) be defined as follows:

\begin{align*} a_1 \amp = \sqrt{3}\\ a_{n+1} \amp =\sqrt{3a_n}, \text{ for } n\geq 1\text{.} \end{align*}

Show that \(\{a_n\}\) converges, and find its limit.