All Questions
12
votes
19answers
7k views
Does .99999… = 1?
I'm told by smart people that 0.999... = 1 and I believe them but is there a proof that explains why?
35
votes
4answers
3k views
Value of $\sum\limits_n x^n$
Why is $\displaystyle \sum\limits_{n=0}^{\infty} 0.7^n$ equal $1/(1-0.7) = 10/3$ ?
Can we generalize the above to
$\displaystyle \sum_{n=0}^{\infty} x^n = \frac{1}{1-x}$ ?
Are there some ...
131
votes
17answers
8k views
Different methods to compute $\sum\limits_{n=1}^\infty \frac{1}{n^2}$
As I have heard people did not trust Euler when he first discovered the formula
$$\zeta(2)=\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}.$$
However, Euler was Euler and he gave other proofs.
I ...
177
votes
8answers
15k views
Is $dy/dx$ not a ratio?
In the book Thomas's Calculus (11th edition) it is mentioned (Section 3.8 pg 225) that the derivative $dy/dx$ is not a ratio. Couldn't it be interpreted as a ratio, because according to the formula ...
3
votes
4answers
8k views
What is 48÷2(9+3)? [duplicate]
Possible Duplicate:
Do values attached to integers have implicit parentheses?
Implicit multiplication order of operation
There is a huge debate on the internet on 48รท2(9+3). I figured if ...
35
votes
13answers
4k views
Proving $\int_{0}^{+\infty} e^{-x^2} dx = \frac{\sqrt \pi}{2}$
How to prove
$$\int_{0}^{+\infty} e^{-x^2} dx = \frac{\sqrt \pi}{2}$$
16
votes
8answers
1k views
How can I evaluate $\sum_{n=0}^\infty (n+1)x^n$
How can if find the sum for:
$$ \sum_{n=1}^\infty \frac{2n}{3^{n+1}} $$
I know the answer thanks to Wolfram Alpha. I'm more concerned with how to get to to that number. It cites tests to prove ...
63
votes
6answers
4k views
How can you prove that a function has no closed form integral?
I've come across statements in the past 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
...
157
votes
13answers
11k views
6
votes
3answers
1k views
How to use the Extended Euclidean Algorithm manually?
I've only found a recursive algorithm of the extended Euclidean algorithm. I'd like to know how to use it by hand. Any idea?
12
votes
4answers
1k views
If $a | m$ and $(a + 1) | m$, prove $a(a + 1) | m$.
Can anyone help me out here? Can't seem to find the right rules of divisibility to show this:
If $a | m$ and $(a + 1) | m$, then $a(a + 1) | m$.
7
votes
2answers
1k views
Given the pdf of independent RVs $I$ and $R$, how to find cdf of $W =I^2R$?
Given pdf of $I$ and $R$ (both $I$ and $R$ are independent RV's), how to find cdf of $W =I^2R$?
Where,
$$
\begin{align}
f_I(i)&=6i(1-i), &0 \leq i \leq 1 \\
f_R(r)&=2r, &0 \leq ...
28
votes
2answers
2k views
The square roots of the primes are linearly independent over the field of rationals
I need to find a way of proving that the square roots of a finite set
of different primes are linearly independent over the field of
rationals. I've tried to solve the problem using elementary ...
20
votes
9answers
3k views
$\sqrt a$ is either an integer or an irrational number.
I got this interesting question in my mind:
How do we prove that if $a \in \mathbb N$, then $\sqrt a$ is an integer or an irrational number?
Can we extend this result? That is, can it be shown ...
130
votes
10answers
28k views
Multiple-choice question about the probability of a random answer to itself being correct
I found this math "problem" on the internet, and I'm wondering if it has an answer:
Question: If you choose an answer to this question at random, what is the probability that you will be correct?
...
54
votes
11answers
3k views
Zero to the zero power - Is $0^0=1$?
Could someone provide me with good explanation of why $0^0 = 1$?
My train of thought:
$x > 0$
$0^x = 0^{x-0} = 0^x/0^0$, so
$0^0 = 0^x/0^x = ?$
Possible answers:
$0^0 * 0^x = 1 * 0^x$, so ...
18
votes
9answers
1k views
$i^2$ why is it $-1$ when you can show it is $1$?
We know $$i^2=-1 $$then why does this happen?
$$
i^2 = \sqrt{-1}\times\sqrt{-1}
$$
$$
=\sqrt{-1\times-1}
$$
$$
=\sqrt{1}
$$
$$
= 1
$$
EDIT: I see this has been dealt with before but at least with ...
49
votes
16answers
8k views
If $AB = I$ then $BA = I$
If $A$ and $B$ are square matrices such that $AB = I$ where $I$ is identity matrix. Show that $BA = I$. I do not understand anything more than the following.
Elementary row operations.
Linear ...
22
votes
5answers
1k views
Limits: How to evaluate $\lim\limits_{x\rightarrow \infty}\sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x$
What methods can be used to evaluate the limit $$\lim_{x\rightarrow\infty} \sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x.$$
In other words, if I am given a polynomial $P(x)=x^n + a_{n-1}x^{n-1} ...
14
votes
19answers
7k views
Proof for formula for sum of sequence $1+2+3+\ldots+n$?
Apparently $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$.
How? What's the proof? Or maybe it is self apparent just looking at the above? Does this problem have a name and maybe a presence on the net? ...
26
votes
8answers
2k views
Division by $0$
I thought it was elementary to me, but I started to do some exercises and came up some definitions I have sort of difficulty to distinguish. In parentheses are my questions.
$x/0$ is Impossible ( ...
46
votes
17answers
4k views
Striking applications of integration by parts
What are your favorite applications of integration by parts?
(The answers can be as lowbrow or highbrow as you wish. I'd just like to get a bunch of these in one place!)
Thanks for your ...
36
votes
3answers
4k views
Norms Induced by Inner Products and the Parallelogram Law
Let $ V $ be a normed vector space (over $\mathbb{R}$, say, for simplicity) with norm $ || \cdot || $.
It's not hard to show that if $|| \cdot || = \sqrt{\langle \cdot, \cdot \rangle}$ for some ...
29
votes
4answers
2k views
Finding the limit of $\frac {n}{\sqrt[n]{n!}}$
I'm trying to find
$$\lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}} .$$
I tried couple of methods: Stolz, Squeeze, D'Alambert
Thanks!
Edit: I can't use Stirling.
11
votes
2answers
947 views
-1 is not 1, so where is the mistake?
I know there must be something unmathematical in the following but I don't know where it is:
\begin{align}
\sqrt{-1} &= i \\ \\
\frac1{\sqrt{-1}} &= \frac1i \\ \\
\frac{\sqrt1}{\sqrt{-1}} ...
31
votes
5answers
3k views
Completion of rational numbers via Cauchy sequences
Can anyone recommend a good self-contained reference for completion of rationals to get reals using Cauchy sequences?
14
votes
2answers
725 views
Possibility to simplify $\sum\limits_{k = - \infty }^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{a + k}} = \frac{\pi }{{\sin \pi a}}} $
Is there any way to show that
$$\sum\limits_{k = - \infty }^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{a + k}} = \frac{1}{a} + \sum\limits_{k = 1}^\infty {{{\left( { - 1} \right)}^k}\left( ...
12
votes
3answers
3k views
Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$
For all $a, m, n \in \mathbb{Z}^+$,
$$\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$$
24
votes
7answers
2k views
Do odd imaginary numbers exist?
Is the concept of an odd imaginary number defined/well-defined/used in mathematics? I searched around but couldn't find anything. Thanks!
3
votes
4answers
1k views
Comaximal ideals in a commutative ring
Let $R$ be a commutative ring and $I_1, \cdots, I_n$ pairwise comaximal ideals in $R$, e.g. $I_i + I_j = R$ for $i \neq j$.
Why are the ideals $I_1^{n_1}, ... , I_r^{n_r}$ (for any $n_1,...,n_r ...
206
votes
33answers
17k views
Do complex numbers really exist?
Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious ...
36
votes
14answers
5k views
Solving the integral $\int_{0}^{\infty} \frac{\sin{x}}{x} \ dx = \frac{\pi}{2}$?
A famous exercise which one encounters while doing Complex Analysis (Residue theory) is to prove that the given integral: $$\int_{0}^{\infty} \frac{\sin{x}}{x} \ dx = \frac{\pi}{2}$$
Well, can anyone ...
44
votes
1answer
2k views
How do we know an $ \aleph_1 $ exists at all?
I have two questions, actually. The first is as the title says: how do we know there exists an infinite cardinal such that there exists no other cardinals between it and $ \aleph_0 $? (We would have ...
27
votes
5answers
1k views
Dominoes and induction, or how does induction work?
I've never really understood why math induction is supposed to work.
You have these 3 steps:
Prove true for base case (n=0 or 1 or whatever)
Assume true for n=k. Call this the induction ...
20
votes
3answers
941 views
Set of continuity points of a real function
I have a question about subsets $$
A \subseteq \mathbb R
$$
for which there exists a function $$f : \mathbb R \to \mathbb R$$ such that the set of continuity points of $f$ is $A$. Can I characterize ...
35
votes
4answers
2k views
“Closed” form for $\sum \frac{1}{n^n}$
Earlier today, I was talking with my friend about some "cool" infinite series and the value they converge to like the Basel problem, Madhava-Leibniz formula for $\pi/4, \log 2$ and similar alternating ...
6
votes
2answers
716 views
bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$
could any one give an example of a bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$?
Thank you.
20
votes
8answers
925 views
why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$
why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$?
I know this is well-known. But how to prove it rigorously? Even mathematical induction does not seem so straight-forward.
...
20
votes
9answers
7k views
How to prove Euler's formula: $e^{it}=\cos t +i\sin t$?
Could you provide a proof of Euler's formula: $e^{it}=\cos t +i\sin t$ ?
thanks.
31
votes
8answers
2k views
How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$?
How can one prove the statement
$$\lim\limits_{x\to 0}\frac{\sin x}x=1$$
without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution.
This is homework. In my ...
10
votes
4answers
711 views
Simpler way to compute a definite integral without resorting to partial fractions?
I found the method of partial fractions very laborious to solve this definite integral :
$$\int_0^\infty \frac{\sqrt[3]{x}}{1 + x^2}\,dx$$
Is there a simpler way to do this ?
46
votes
17answers
2k views
Proving the identity $\sum\limits_{k=1}^n {k^3} = \left(\sum\limits_{k=1}^n k\right)^2$ without induction
I recently proved that
$$
\sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k \right)^2
$$
Using mathematical induction. I'm interested if there's an intuitive explanation, or even a combinatorial ...
48
votes
5answers
3k views
Is there an elementary proof that $\sum \limits_{k=1}^n \frac1k$ is never an integer?
If $n>1$ is an integer, then $\sum \limits_{k=1}^n \frac1k$ is not an integer.
If you know Bertrand's Postulate, then you know there must be a prime $p$ between $n/2$ and $n$, so $\frac 1p$ ...
28
votes
2answers
897 views
Evaluating $\int P(\sin x, \cos x) \text{d}x$
Suppose $\displaystyle P(x,y)$ a polynomial in the variables $x,y$.
For example, $\displaystyle x^4$ or $\displaystyle x^3y^2 + 3xy + 1$.
Is there a general method which allows us to evaluate the ...
37
votes
3answers
2k views
Why is $1^{\infty}$ considered to be an indeterminate form
From Wikipedia: In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are ...
9
votes
4answers
1k views
Sine function dense in $[-1,1]$
We know that the sine function takes it values between $[-1,1]$. So is the set $$A = \{ \sin{n} \ : \ n \in \mathbb{N}\}$$ dense in $[-1,1]$. Generally, for showing the set is dense, one proceeds, by ...
29
votes
2answers
1k views
Identity for convolution of central binomial coefficients
It's not difficult to show that
$$(1-z^2)^{-1/2}=\sum_{n=0}^\infty \binom{2n}{n}2^{-2n}z^{2n}$$
On the other hand, we have $(1-z^2)^{-1}=\sum z^{2n}$. Squaring the first power series and comparing ...
13
votes
3answers
648 views
Universal Chord Theorem
Let $f \in C[0,1]$ and $f(0)=f(1)$.
How do we prove $\exists a \in [0,1/2]$ such that $f(a)=f(a+1/2)$?
In fact, for every positive integer $n$, there is some $a$, such that $f(a) = ...
12
votes
3answers
1k views
Is there a name for function with the exponential property $f(x+y)=f(x) \times f(y)$?
I was wondering if there is a name for a function that satisfies the conditions
$f:\mathbb{R} \to \mathbb{R}$ and $f(x+y)=f(x) \times f(y)$?
Thanks and regards!
4
votes
1answer
745 views
Finite Sum of Power?
Sorry for the remedial math but:
Can someone tell me how to get a closed form for
$$\sum_{k=1}^n k^x$$
For $x = 1$, it's just the classic $n(n+1)/2$. What is it for $x > 1$?
