Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Infinity wikipedia, lookup

Functional decomposition wikipedia, lookup

Georg Cantor's first set theory article wikipedia, lookup

Non-standard calculus wikipedia, lookup

Karhunen–Loève theorem wikipedia, lookup

Uniform convergence wikipedia, lookup

Fundamental theorem of algebra wikipedia, lookup

Law of large numbers wikipedia, lookup

Large numbers wikipedia, lookup

Non-standard analysis wikipedia, lookup

Proofs of Fermat's little theorem wikipedia, lookup

Collatz conjecture wikipedia, lookup

Hyperreal number wikipedia, lookup

Sequence wikipedia, lookup

Transcript
```Sequences (11/13/13)


A sequence is an infinite list of real numbers:
{a1, a2, a3, a4, a5 ….} = {an}. (Order counts!)
A sequence can be described by
–
–

listing some initial terms, hence establishing a
pattern, or
giving its general formula.
Examples: {1, 4, 9, 16,…} = formula?
{n / 2n} = what initial elements?
Clicker Question 1

What are the first 4 terms of {(-1)n+1 / n} ?
– A. {1, 1/2, 1/3, 1/4,…}
– B. {1, -1/2, 1/3, -1/4,…}
– C. {-1, 1/2, -1/3, 1/4,…}
– D. {-1, -1/2, -1/3, -1/4,…}
– E. {1, -2, 3, -4,…}
Clicker Question 2

What is a formula for the sequence
{2/3, 4/9, 6/27, 8/81,…} ?
– A. {2n / 3n}
– B. {2n / 3n}
– C. {2n / 3n}
– D. {2n / 3n}
– E. {(n + 1)/3n}
Convergent and Divergent
Sequences



A sequence {an} converges to L (a real
number) if, given any positive distance from
L, we can go far enough out in the sequence
so that every term from there out is within
that given distance from L.
We write limnan = L or {an}  L .
If {an} does not converge, we say it diverges.
Examples






Do these sequences converge or diverge? If
converge, to what? (Btw, sound familiar?)
{1, 1/4, 1/9, 1/16, …}
{(-1)n+1(1/n2)}
{(-1)n+1}
{1, 1+1/4, 1+1/4+1/9, 1+1/4+1/9+1/16, …}
{1, 1+1/2, 1+1/2+1/3, 1+1/2+1/3+1/4, …}
Clicker Question 3

{1, 1+1/2, 1+1/2+1/4, 1+1/2+1/4+1/8,…}
– A. converges to 1.
– B. converges to 2.
– C. converges to some number greater
than 2.
– D. diverges
Monotone & Bounded Sequences






If a1 < a2 < a3 < …, {an} is called increasing.
Likewise decreasing.
In either case, {an} is called monotone.
If an < M (for some M and for all n),
{an} is called bounded above.
Likewise bounded below.
If a sequence is monotone, we can simply
say bounded.
Monotone Convergence

Theorem: If {an} is both monotone and
bounded, then it is convergent.

Example:
{1, 1+1/4, 1+1/4+1/9, 1+1/4+1/9+1/16,
…}

is bounded above by 1 + 1 1 2 dx = ??
x
Hence it must converge, but to what??
For Friday


Read Section 11.1 as needed (long section).
In that section, please do Exercises 1, 2,
3-17 odd, 23, 25, 27, 31, 33, 35, 43, 49, 53,
55, 65, 73, 75, 77. (Again, stay calm, most of
these are quickies!)
```
Related documents