Sequences - Revision history

Lfox: Changed redirect target from Pagename to Sequence

2024-01-21T01:55:52Z

Changed redirect target from Pagename to Sequence

← Older revision		Revision as of 01:55, 21 January 2024
Line 1:		Line 1:
	#REDIRECT [[~~pagename~~]]		#REDIRECT [[Sequence]]

Lfox: Redirected page to Pagename

2024-01-21T01:55:41Z

Redirected page to Pagename

← Older revision		Revision as of 01:55, 21 January 2024
Line 1:		Line 1:
	~~A '''sequence''' is a~~ [[~~function~~]] ~~which takes [[natural numbers]] as inputs, and returns [[numbers]] as outputs.~~		#REDIRECT [[pagename]]

	~~== The Cauchy condition ==~~
	~~Suppose that <math>\{ a_n \}_{n=1}^\infty</math> is a sequence that computes [[π]].~~

	The practical question to answer, for someone computing this sequence on a computer, is "after how many steps can I stop running the computer program?" The answer is that you should wait until successive steps of the sequence vary by amounts smaller than your desired accuracy.

	~~For example, if your target accuracy is 0.01, and if the printout looks like~~
	~~{\| class="wikitable"~~
	\|+
	~~!<math>n</math>~~
	~~!<math>a_n</math>~~
	\|-
	\|1
	\|4
	\|-
	\|2
	~~\|3.0~~
	\|-
	\|3
	~~\|3.2~~
	\|-
	\|4
	~~\|3.09~~
	\|-
	\|5
	~~\|3.16~~
	\|-
	\|6
	~~\|3.139~~
	\|-
	\|7
	~~\|3.143~~
	\|-
	\|8
	~~\|3.1408~~
	\|-
	\|9
	~~\|3.1419~~
	\|-
	~~\|10~~
	~~\|3.14149~~
	\|-
	~~\|...~~
	~~\|...~~
	\|}
	~~then we can stop computing around step 9, because at that point the output of the programming is varying by amounts much smaller than 0.01.~~

	What I have done in the above paragraph is I have inadvertently assumed that the sequence satisfies the '''Cauchy condition'''. A sequence is said to satisfy the Cauchy condition if, for any level of desired accuracy, there is some point beyond which any two outputs of the algorithm differ by an amount smaller than the desired accuracy. More formally, <math>\{a_n\}_{n=1}^\infty</math> satisfies the Cauchy condition if for any <math>\epsilon : \mathbb{Q}</math> such that <math>\epsilon > 0, </math> there exists an <math>N : \mathbb{N}</math> such that if <math>n,m \geq N</math> then <math>\|a_n - a_m \| \leq \epsilon</math>.

	~~== Examples ==~~
	~~The sequence <math>\{2n+1\}_{n=0}^\infty</math> could be properly thought of as that concept which---in appropriate contexts---subsumes the following concretes:~~
	* the method that a child uses to produce the ''n''th odd number
	* a C++ program which takes in a uint32, ''n'', and outputs 2''n''+1
	* the following table on my screen [a screenshot of a few rows of an excel table where the entries are 2*row + 1 ]
	* etc.

Lfox at 06:29, 20 January 2024

2024-01-20T06:29:41Z

← Older revision		Revision as of 06:29, 20 January 2024
Line 1:		Line 1:
	A '''sequence''' is a [[function]] which takes [[~~Natural number\|~~natural numbers]] as inputs, and returns [[numbers]] as outputs.		A '''sequence''' is a [[function]] which takes [[natural numbers]] as inputs, and returns [[numbers]] as outputs.

	== The Cauchy condition ==		== The Cauchy condition ==

Lfox: /* The Cauchy condition */

2024-01-20T06:29:10Z

The Cauchy condition

← Older revision		Revision as of 06:29, 20 January 2024
Line 47:		Line 47:
	then we can stop computing around step 9, because at that point the output of the programming is varying by amounts much smaller than 0.01.		then we can stop computing around step 9, because at that point the output of the programming is varying by amounts much smaller than 0.01.

	What I have done in the above paragraph is I have inadvertently assumed that the sequence satisfies the '''Cauchy condition'''. A sequence is said to satisfy the Cauchy condition if, for any level of desired accuracy, there is some point beyond which any two outputs of the algorithm differ by an amount smaller than the desired accuracy. More formally, <math>\{a_n\}_{n=1}^\infty</math> satisfies the Cauchy condition if for [[any]] <math>\epsilon : \mathbb{Q}</math> such that <math>\epsilon > 0, </math> there exists an <math>N : \mathbb{N}</math> such that if <math>n,m \geq N</math> then <math>\|a_n - a_m \| \leq \epsilon</math>.		What I have done in the above paragraph is I have inadvertently assumed that the sequence satisfies the '''Cauchy condition'''. A sequence is said to satisfy the Cauchy condition if, for any level of desired accuracy, there is some point beyond which any two outputs of the algorithm differ by an amount smaller than the desired accuracy. More formally, <math>\{a_n\}_{n=1}^\infty</math> satisfies the Cauchy condition if for any <math>\epsilon : \mathbb{Q}</math> such that <math>\epsilon > 0, </math> there exists an <math>N : \mathbb{N}</math> such that if <math>n,m \geq N</math> then <math>\|a_n - a_m \| \leq \epsilon</math>.

	== Examples ==		== Examples ==

Lfox at 06:10, 17 January 2024

2024-01-17T06:10:40Z

@@ Line 1: / Line 1: @@
-From my email to RS<blockquote> I have noticed that there are many specific "infinite sets" that I think are---or rather, are philosophically sloppy stand-ins for---valid concepts. For example, the infinite (ordered) set <math>\{2n+1\}_{n=0}^\infty</math> could be properly thought of as that concept which---in appropriate contexts---subsumes the following concretes:
+A '''sequence''' is a [[function]] which takes [[Natural number|natural numbers]] as inputs, and returns [[numbers]] as outputs.
 * the method that a child uses to produce the ''n''th odd number
 * a C++ program which takes in a uint32, ''n'', and outputs 2''n''+1
 * the following table on my screen [a screenshot of a few rows of an excel table where the entries are 2*row + 1 ]
 * etc.

Lfox: created page

2024-01-16T00:16:56Z

created page

New page

From my email to RS<blockquote> I have noticed that there are many specific "infinite sets" that I think are---or rather, are philosophically sloppy stand-ins for---valid concepts. For example, the infinite (ordered) set <math>\{2n+1\}_{n=0}^\infty</math> could be properly thought of as that concept which---in appropriate contexts---subsumes the following concretes:

* the method that a child uses to produce the ''n''th odd number
* a C++ program which takes in a uint32, ''n'', and outputs 2''n''+1
* the following table on my screen [a screenshot of a few rows of an excel table where the entries are 2*row + 1 ]
* etc.

Or the infinite set <math>\mathbb{N}</math> is just a stand-in for the concept of natural numbers, which subsumes the concepts 1, 2, 3, etc, which in turn subsume concretes like 1 apple, 2 oranges, 3 bananas, etc.</blockquote>