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Questions tagged [faq]

This is meant for questions which are generalized forms of questions which get asked frequently. See tag details for more information.

When you add a question marked faq, please also update the list of questions: List of Generalizations of Common Questions

The question which prompted this: Coping with *abstract* duplicate questions.

Note: Even though one might argue that tagging a question as faq should be enough, and there is no need to update the above list, updating the above list will serve to bring this policy back to attention and help raise awareness periodically.

123 questions
881
votes
59 answers

Different ways to prove $\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}$ (the Basel problem)

As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem) $$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}$$ However, Euler was Euler and he gave other proofs. I believe many of you…
AD - Stop Putin -
  • 11,200
456
votes
24 answers

How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$?

How can I evaluate $$\sum_{n=1}^\infty\frac{2n}{3^{n+1}}$$? I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is convergent, but my class has never learned these…
backus
  • 4,793
379
votes
24 answers

Zero to the zero power – is $0^0=1$?

Could someone provide me with a good explanation of why $0^0=1$? My train of thought: $$x>0\\ 0^x=0^{x-0}=\frac{0^x}{0^0}$$ so $$0^0=\frac{0^x}{0^x}=\,?$$ Possible answers: $0^0\cdot0^x=1\cdot0^0$, so $0^0=1$ $0^0=\frac{0^x}{0^x}=\frac00$, which is…
Stas
  • 4,049
369
votes
7 answers

How can you prove that a function has no closed form integral?

In the past, I've come across statements along the lines of "function $f(x)$ has no closed form integral", which I assume means that there is no combination of the operations: addition/subtraction multiplication/division raising to powers and…
Simon Nickerson
  • 5,791
334
votes
17 answers

Any open subset of $\Bbb R$ is a countable union of disjoint open intervals

Let $U$ be an open set in $\mathbb R$. Then $U$ is a countable union of disjoint intervals. This question has probably been asked. However, I am not interested in just getting the answer to it. Rather, I am interested in collecting as many…
Orest Xherija
  • 1,079
222
votes
4 answers

Overview of basic results about images and preimages

Are there some good overviews of basic facts about images and inverse images of sets under functions?
Martin Sleziak
  • 56,060
202
votes
9 answers

What Does it Really Mean to Have Different Kinds of Infinities?

Can someone explain to me how there can be different kinds of infinities? I was reading "The man who loved only numbers" by Paul Hoffman and came across the concept of countable and uncountable infinities, but they're only words to me. Any help…
Allain Lalonde
  • 2,147
197
votes
13 answers

Convergence of the series $\sum\limits_{n=1}^{\infty} \frac1{n^p}$ for $p > 1$

I am trying to prove the convergence of the $p$-series $$\sum_{n=1}^{\infty} \frac1{n^p}.$$ for $p > 1$. I am wondering if there is a proof that this series converges, either directly or by applying some test for convergence.
admchrch
  • 2,932
195
votes
14 answers

Intuition behind Matrix Multiplication

If I multiply two numbers, say $3$ and $5$, I know it means add $3$ to itself $5$ times or add $5$ to itself $3$ times. But If I multiply two matrices, what does it mean ? I mean I can't think it in terms of repetitive addition. What is the…
Happy Mittal
  • 3,307
183
votes
14 answers

How to prove: if $a,b \in \mathbb N$, then $a^{1/b}$ is an integer or an irrational number?

It is well known that $\sqrt{2}$ is irrational, and by modifying the proof (replacing even with divisible by $3$), one can prove that $\sqrt{3}$ is irrational, as well. On the other hand, clearly $\sqrt{n^2} = n$ for any positive integer $n$. It…
anonymous
167
votes
1 answer

Overview of basic facts about Cauchy functional equation

The Cauchy functional equation asks about functions $f \colon \mathbb R \to \mathbb R$ such that $$f(x+y)=f(x)+f(y).$$ It is a very well-known functional equation, which appears in various areas of mathematics ranging from exercises in freshman…
Martin Sleziak
  • 56,060
162
votes
21 answers

Best book of topology for beginner?

I am a graduate student of math right now but I was not able to get a topology subject in my undergrad... I just would like to know if you guys know the best one..
gg1
  • 111
155
votes
41 answers

Why is negative times negative = positive?

Someone recently asked me why a negative $\times$ a negative is positive, and why a negative $\times$ a positive is negative, etc. I went ahead and gave them a proof by contradiction like this: Assume $(-x) \cdot (-y) = -xy$ Then divide both sides…
Sev
  • 2,151
154
votes
8 answers

Intuition behind Conditional Expectation

I'm struggling with the concept of conditional expectation. First of all, if you have a link to any explanation that goes beyond showing that it is a generalization of elementary intuitive concepts, please let me know. Let me get more specific. Let…
Stefan
  • 7,185
141
votes
36 answers

Proof that $1+2+3+4+\cdots+n = \frac{n\times(n+1)}2$

Why is $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$ $\space$ ?
b1_
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