Sean Fitzpatrick |
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University of Lethbridge |
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Write out the first 5 terms in each sequence:
\(\di a_n = \frac{2^n}{n}\)
\(\di b_n = \frac{(-1)^n n^2}{n!}\)
\(\di c_n = n\cos(\pi n)\)
Determine a formula for the sequence:
\(2, 4, 8, 16, 32, \ldots\)
\(\di 1,-\frac23,\frac45,-\frac67,\frac89,\ldots\)
\(\di \frac{1}{3}, \frac{3}{8}, \frac{1}{3}, \frac{7}{24}, \frac{9}{35},\ldots\)
What does it mean to say that a sequence \(\{a_n\}\text{:}\)
converges?
is monotonic?
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 \(n\geq 1\text{,}\) \(a_{n+2}=a_n+a_{n+1}\text{.}\)
Arithmetic: \(a_1=b\text{,}\) \(a_{n+1}=a_n+d\)
Geometric: \(a_1 = k\text{,}\) \(a_{n+1}=ra_n\)
Factorial: \(a_0=1\text{,}\) \(a_{n}=na_{n-1}\)
And so many more
Let \(\{a_n\}\) be a sequence. We say that the sequence converges to a limit \(L\text{,}\) and write
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
Find the limit of the sequence:
\(\di a_n = \frac1n\) (Assumes the Archimedian property of \(\R\text{.}\))
\(\di a_n = r^n\text{,}\) if:
\(r\gt 1\)
\(r=1\)
\(0\lt r \lt 1\)
\(\di a_n = \frac{2+4n-5n^2}{2n^2+9n}\)
Keep finding those limits:
\(a_n = \cos(\pi n/3)\)
\(a\di _n = \frac{n}{n+1}-\frac{n+1}{n}\)
\(\di a_n = \left(1+\frac{3}{n}\right)^n\)
\(\di a_n = \frac{2n}{\sqrt{1+9n^2}}\)
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, \(\lim\limits_{n\to \infty}f(a_n)=f\left(\lim\limits_{n\to\infty}a_n\right)\text{.}\)
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 \(n\text{.}\)
Any convergent sequence is bounded.
Any bounded, monotone sequence converges.
Of the following sequences, which are bounded? Which are montone?
\(\di a_n = \frac{5}{2n+1}\)
\(\di a_n = \frac{2n-1}{3n+5}\)
\(\di a_n = \sin(n\pi/4)\)
\(\di a_n = \frac{n^2-4}{3n}\)
Let \(\{a_n\}\) be defined as follows:
Show that \(\{a_n\}\) converges, and find its limit.