Notes on sequences.

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notes/_journal/2024-03-01.md

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+---
+title: "2024-03-01"
+---
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notes/algebra/sequences.md

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+---
+title: Sequences
+TARGET DECK: Obsidian::STEM
+FILE TAGS: algebra
+tags:
+  - algebra
+---
+
+## Overview
+
+A **sequence** is an ordered list of numbers. These are usually represented as a **closed formula** or a **recursive definition**.
+
+%%ANKI
+Basic
+What is a sequence?
+Back: An ordered list of values.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1709302343193-->
+END%%
+
+%%ANKI
+Basic
+How does sequence $0, 1, 2, \ldots$ differ from set $\mathbb{N}$?
+Back: Order matters in the former.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1709302343198-->
+END%%
+
+%%ANKI
+Basic
+How is sequence $a_1, a_2, \ldots$ denoted?
+Back: $(a_n)_{n \geq 1}$
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1709302343201-->
+END%%
+
+%%ANKI
+Basic
+What is the first index of $(a_n)_{n \geq 1}$?
+Back: $1$
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1709302343204-->
+END%%
+
+%%ANKI
+Basic
+What terms make up the expanded form of sequence $(a_n)_{n \geq 0}$?
+Back: $a_0, a_1, a_2, \ldots$
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1709302343207-->
+END%%
+
+%%ANKI
+Basic
+What term refers to the subscripts in sequence $a_0, a_1, a_2, \ldots$?
+Back: Indices.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1709302343211-->
+END%%
+
+%%ANKI
+Basic
+How do functions and sequences relate to one another?
+Back: Every sequence is equivalent to a function mapping index to value.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1709302343214-->
+END%%
+
+%%ANKI
+Basic
+How is sequence $a_0, a_1, a_2, \ldots$ interpreted as a function?
+Back: As a function that assigns $n \mapsto a_n$.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1709302343216-->
+END%%
+
+%%ANKI
+Basic
+What two methods allow exact specification of a sequence?
+Back: Closed formulas and recursive definitions.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1709302343220-->
+END%%
+
+%%ANKI
+Basic
+What is a closed formula for sequence e.g. $(a_n)_{n \geq 0}$?
+Back: A formula that can be computed using a fixed number of "well-known" standard operations, independent of $n$.
+Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
+<!--ID: 1709302343222-->
+END%%
+
+%%ANKI
+Cloze
+{$n!$} is a closed formula even though equivalent expression {$\Pi_{k=1}^n k$} is not.
+Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
+<!--ID: 1709302343225-->
+END%%
+
+%%ANKI
+Basic
+Why is $\Pi_{k=1}^n k$ not considered a closed formula?
+Back: The number of "well-known" standard operations (i.e. multiplications) scale with $n$.
+Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
+<!--ID: 1709302343228-->
+END%%
+
+%%ANKI
+Basic
+Why is $n!$ considered a closed formula?
+Back: Because the factorial operation is considered a "well-known" standard operation.
+Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
+<!--ID: 1709302343231-->
+END%%
+
+%%ANKI
+Basic
+What two parts make up a recursive definition of a sequence?
+Back: The recurrence relation and an initial condition.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1709302343234-->
+END%%
+
+%%ANKI
+Basic
+What distinguishes a recurrence relation and a recursive definition?
+Back: A recursive definition includes a recurrence relation along with an initial condition.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1709302343238-->
+END%%
+
+%%ANKI
+Basic
+How is sequence $(a_n)_{n \geq 0} = 1, 2, 4, 8, \ldots$ represented as a closed formula?
+Back: $a_n = 2^n$
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1709302343242-->
+END%%
+
+%%ANKI
+Basic
+How is sequence $(a_n)_{n \geq 0} = 1, 2, 4, 8, \ldots$ represented as a recursive definition?
+Back: $a_n = 2 \cdot a_{n-1}$ with $a_0 = 1$.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1709302343245-->
+END%%
+
+%%ANKI
+Cloze
+It is harder to find {terms of a sequence} with a recursive definition than closed formula.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1709302343249-->
+END%%
+
+%%ANKI
+Basic
+Which is considered easier to find: closed formulas or recursive definitions?
+Back: Recursive definitions.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1709302343252-->
+END%%
+
+%%ANKI
+Basic
+What general strategy is typically employed when finding a closed formula for a sequence?
+Back: First find a recursive definition.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1709302343255-->
+END%%
+
+## References
+
+* Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+* Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).

notes/algorithms/sorting/merge-sort.md

@@ -71,7 +71,7 @@ END%%
 %%ANKI
 Basic
 What is the "combine" step of merge sort?
-Back: Merge the subarrays sorted after the "conquer" step.
+Back: Merge the sorted subarrays produced by the "conquer" step.
 Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
 <!--ID: 1708742467125-->
 END%%

notes/logic/equiv-trans.md

@@ -707,7 +707,7 @@ END%%
 
 %%ANKI
 Basic
-What is the role of $\bar{e}$ role in textual substitution $E_{\bar{e}}^{\bar{x}}$?
+What is the role of $\bar{e}$ in textual substitution $E_{\bar{e}}^{\bar{x}}$?
 Back: It is the expressions that are substituted into $E$.
 Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
 <!--ID: 1707762304127-->