List of Generalizations of Common Questions

Question

Bill Dubuque raised an excellent point here: Coping with *abstract* duplicate questions.

I suggest we use this question as a list of the generalized questions we create.

I suggest we categorize based on topic (please edit the question). Also please feel free to suggest a better way to list these.

Also, as per Jeff's recommendation, please tag these questions as faq.

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Arithmetic arithmetic

• Laws of signs (minus times minus is plus): Why negative times negative = positive?

• Order of operations in arithmetic: What is the standard interpretation of order of operations for the basic arithmetic operations?

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Algebra/Precalculus algebra-precalculus

• Geometric Series: Value of $\sum\limits_n x^n$

• Solving equations with multiple absolute values: What is the best way to solve an equation involving multiple absolute values?

• Extraneous solutions to equations with a square root: Is there a name for this strange solution to a quadratic equation involving a square root?

• Principal $n$-th roots:

  • Significance of $\displaystyle\sqrt[n]{a^n} $?
  • Why is the even root of a number always positive?

• $0! = 1$: Prove $0! = 1$ from first principles

• Triangular numbers:

  • What is the term for a factorial type operation, but with summation instead of products?
  • Proof for formula for sum of sequence $1+2+3+\ldots+n$?

• Pyramidal numbers:

  • How do I come up with a function to count a pyramid of apples?
  • Proof that $\sum\limits_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$?
  • How to get to the formula for the sum of squares of first n numbers?
  • how can one find the value of the expression, $(1^2+2^2+3^2+\cdots+n^2)$
  • How to calculate the sum of $(n-1)^2+(n-2)^2+...+1$?

• Proving the Identity $\sum\limits_{k=1}^n {k^3} = \left(\sum\limits_{k=1}^n k\right)^2$

  • With Induction: Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using induction
  • Without Induction: Proving the identity $\sum\limits_{k=1}^n {k^3} = \left(\sum\limits_{k=1}^n k\right)^2$ without induction

• Partial fraction decomposition of rational functions: Converting multiplying fractions to sum of fractions

• Highest power of a prime $p$ dividing $N!$, number of zeros at the end of $N!$ and related questions: Highest power of a prime $p$ dividing $N!$

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Calculus calculus

• Integrating a polynomial in $\sin x$ and $\cos x$: Evaluating $\int P(\sin x, \cos x) \text{d}x$

• Integration using partial fractions: Integration by partial fractions; how and why does it work?

• Intuitive meaning of Euler's constant $e$: Intuitive Understanding of the constant "e"

• Evaluating limits of the form $\lim_{x\to \infty} P(x)^{1/n}-x$ where $P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0$ is a monic polynomial: Limits: How to evaluate $\lim\limits_{x\rightarrow \infty}\sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x$

• Finding the limit of rational functions at infinity: Finding the limit of $\frac{Q(n)}{P(n)}$ where $Q,P$ are polynomials

• $\zeta(2)$: Different methods to compute $\sum\limits_{n=1}^\infty \frac{1}{n^2}$

• Divergence of the harmonic series: Why does the series $\frac 1 1 + \frac 12 + \frac 13 + \cdots$ not converge?

• Universal Chord Theorem: Universal Chord Theorem

• Nested radical series: $\sqrt{c+\sqrt{c+\sqrt{c+\cdots}}}$, or the limit of the sequence $x_{n+1} = \sqrt{c+x_n}$

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Exponentiation exponentiation

Solving $x^x=y$ for $x$: Is $x^x=y$ solvable for $x$?

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Functional equations functional-equations

$f(x+y)=f(x)f(y)$: Is there a name for function with the exponential property $f(x+y)=f(x) \times f(y)$?

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Linear algebra linear-algebra

• Definition of Matrix Multiplication: (Maybe there should just be one canonical one?)
  • What does matrix multiplication actually mean?
  • What does matrix multiplication have to do with scalar multiplication?
• On the determinant:
  • What's an intuitive way to think about the determinant?

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Number Theory elementary-number-theory

• Modular exponentiation: How do I compute $a^b\,\bmod c$ by hand?

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Probability probability

• "Two Children Puzzle / Boy Born on a Tuesday" and variants:

  • In a family with two children, what are the chances, if one of the children is a girl, that both children are girls?
  • New Two Children problem
  • Boy Born on a Tuesday - is it just a language trick?

• A "liar paradox" variant: Multiple-choice question about the probability of a random answer to itself being correct

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Sequences/Series sequences-and-series (also summation summation)

• Geometric Series: Value of $\sum\limits_n x^n$

• Finding the limit of rational functions at infinity: Finding the limit of $\frac{Q(n)}{P(n)}$ where $Q,P$ are polynomials

• $\zeta(2)$: Different methods to compute $\sum\limits_{n=1}^\infty \frac{1}{n^2}$

• Divergence of the harmonic series: Why does the series $\frac 1 1 + \frac 12 + \frac 13 + \cdots$ not converge?

• Nested radical series: $\sqrt{c+\sqrt{c+\sqrt{c+\cdots}}}$, or the limit of the sequence $x_{n+1} = \sqrt{c+x_n}$

• Limit of exponential sequence and $n$ factorial: Prove that $\lim \limits_{n \to \infty} \frac{x^n}{n!} = 0$, $x \in \Bbb R$.

• Faulhaber's formula / sum of powers Closed form for $\sum\limits_{k=1}^n k^x$: Finite Sum of Power?

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Statistics statistics

• Upper tail inequality for the standard normal distribution:
  • Proof of upper-tail inequality for standard normal distribution,
  • Proof of an estimate for the tail of a normal distribution

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Topology topology

• Book recommendations:
  • best book for topology?
  • Can anybody recommend me a topology textbook?
  • Choosing a topology text
  • Introductory book on Topology
  • Reference for general-topology (Related, but not the same)

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Trigonometry trigonometry

• (Confusing) notation for inverse functions ($\sin^{-1}$ vs. $\arcsin$): $\arcsin$ written as $\sin^{-1}(x)$

Answer 1

This "answer" is simply suggesting another category (Logic? Proof-writing?), perhaps each a separate category) for frequently asked questions about

Logic:

• the meaning(s) of implication (and understanding it, intuitively),
• understanding quantifiers in predicate logic; ($\exists$ vs. $\forall$),
• equivalence between an implication and its contrapositive,
• understanding what is meant by a tautolology,...a contradition...etc.

Proof writing:

• understanding the difference between proving the contrapositive of an implication and proof by contradiction.

• understanding the "logic"/and or intuition of proof by induction, and when it is appropriate to use.

• Proof by induction: How to prove $P(n)$ by induction on n.

This "post" is merely a rough draft. I can certainly work at it, updating to include representative links, and revise this answer for inclusion as a category/categories in the original post, at top.

But I invite anyone noticing the frequency of such question and/or conceptual confusions to suggest links to representative posts, with an eye for the quality of ensuing answers in those post. Also feel free to include any other frequently encountered questions tagged "logic" and/or "proof-writing"/proof-strategies that I have not yet addressed in this post.

Answer 2

Is the goal to find examples "in the wild" of common questions, that we then coopt and retag as faq? Or do we create new community wiki questions tagged faq specifically for the purpose of answering them comprehensively (as was done on SO)?

Answer 3

The Lindemann–Weierstrass theorem can be used to prove many numbers transcendental, as in

• Transcendentality of the $\log$ of the golden mean
• Prove that $\pi$ is a transcendental number
• How hard is the proof of $\pi$ or e being transcendental?
• Is ln(n) transcendental for all integer $n > 1$?

There are some other relevant theorems, like Deciding whether $2^{\sqrt2}$ is irrational/transcendental.

Answer 4

My suggestion is that there is an "examples" tag which can be used to link/merge examples of general questions into the most general question which has been posted on a topic.