All Questions
151,818
questions
427
votes
23
answers
66k
views
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 ...
- 4,485
98
votes
9
answers
9k
views
Divisibility by 7 rule, and Congruence Arithmetic Laws
I have seen other criteria for divisibility by 7. Criterion described below present in the book Handbook of Mathematics for IN Bronshtein (p. 323) is interesting, but could not prove it.
Let $n = (...
- 5,087
124
votes
12
answers
20k
views
Modular exponentiation by hand ($a^b\bmod c$)
How do I efficiently compute $a^b\bmod c$:
When $b$ is huge, for instance $5^{844325}\bmod 21$?
When $b$ is less than $c$ but it would still be a lot of work to multiply $a$ by itself $b$ times, for ...
- 45.8k
519
votes
27
answers
192k
views
How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$?
How can one prove the statement
$$\lim_{x\to 0}\frac{\sin x}x=1$$
without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution.
This is homework. In my math ...
- 8,619
174
votes
8
answers
81k
views
How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression?
How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression? For example here is the sum of $\cos$ series:
$$\sum_{k=0}^{n-1}\cos (a+k \cdot d) =\frac{\sin(n \times \frac{...
- 21.4k
7
votes
4
answers
5k
views
Mod of numbers with large exponents [modular order reduction]
I've read about Fermat's little theorem and generally how congruence works. But I can't figure out how to work out these two:
$13^{100} \bmod 7$
$7^{100} \bmod 13$
I've also heard of this formula:
$...
- 221
1205
votes
26
answers
125k
views
Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
In Thomas's Calculus (11th edition), it is mentioned (Section 3.8 pg 225) that the derivative $dy/dx$ is not a ratio. Couldn't it be interpreted as a ratio, because according to the formula $dy = f'(x)...
- 15k
180
votes
14
answers
16k
views
Why $\sqrt{-1 \cdot {-1}} \neq \sqrt{-1}^2$?
I know there must be something unmathematical in the following but I don't know where it is:
\begin{align}
\sqrt{-1} &= i \\\\\
\frac1{\sqrt{-1}} &= \frac1i \\\\
\frac{\sqrt1}{\sqrt{-1}} &...
- 2,003
369
votes
33
answers
112k
views
If $AB = I$ then $BA = I$
If $A$ and $B$ are square matrices such that $AB = I$, where $I$ is the identity matrix, show that $BA = I$.
I do not understand anything more than the following.
Elementary row operations.
Linear ...
- 5,737
789
votes
49
answers
123k
views
Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$ (Basel problem)
As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem)
$$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}.$$
However, Euler was Euler ...
167
votes
9
answers
42k
views
Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$
For all $a, m, n \in \mathbb{Z}^+$,
$$\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$$
Juan Liner
84
votes
5
answers
30k
views
How to use the Extended Euclidean Algorithm manually?
I've only found a recursive algorithm of the extended Euclidean algorithm. I'd like to know how to use it by hand. Any idea?
- 2,247
57
votes
15
answers
18k
views
How to prove that $\log(x)<x$ when $x>1$?
It's very basic but I'm having trouble to find a way to prove this inequality
$\log(x)<x$
when $x>1$
($\log(x)$ is the natural logarithm)
I can think about the two graphs but I can't find ...
- 2,327
82
votes
5
answers
20k
views
$\sqrt{c+\sqrt{c+\sqrt{c+\cdots}}}$, or the limit of the sequence $x_{n+1} = \sqrt{c+x_n}$
(Fitzpatrick Advanced Calculus 2e, Sec. 2.4 #12)
For $c \gt 0$, consider the quadratic equation
$x^2 - x - c = 0, x > 0$.
Define the sequence $\{x_n\}$ recursively by fixing $|x_1| \lt c$ and ...
- 965
114
votes
3
answers
8k
views
Are all limits solvable without L'Hôpital Rule or Series Expansion
Is it always possible to find the limit of a function without using L'Hôpital Rule or Series Expansion?
For example,
$$\lim_{x\to0}\frac{\tan x-x}{x^3}$$
$$\lim_{x\to0}\frac{\sin x-x}{x^3}$$
$$\...
338
votes
30
answers
54k
views
Is it true that $0.999999999\ldots=1$?
I'm told by smart people that
$$0.999999999\ldots=1$$
and I believe them, but is there a proof that explains why this is?
69
votes
18
answers
18k
views
Proof of the hockey stick/Zhu Shijie identity $\sum\limits_{t=0}^n \binom tk = \binom{n+1}{k+1}$
After reading this question, the most popular answer use the identity
$$\sum_{t=0}^n \binom{t}{k} = \binom{n+1}{k+1}.$$
What's the name of this identity? Is it the identity of the Pascal's triangle ...
- 1,530
22
votes
5
answers
7k
views
Solving linear congruences by hand: modular fractions and inverses
When I am faced with a simple linear congruence such as
$$9x \equiv 7 \pmod{13}$$
and I am working without any calculating aid handy, I tend to do something like the following:
"Notice" that adding $...
- 19.2k
444
votes
15
answers
58k
views
Why does $1+2+3+\cdots = -\frac{1}{12}$?
$\displaystyle\sum_{n=1}^\infty \frac{1}{n^s}$ only converges to $\zeta(s)$ if $\text{Re}(s) > 1$.
Why should analytically continuing to $\zeta(-1)$ give the right answer?
perplexed
130
votes
1
answer
24k
views
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 ...
67
votes
2
answers
9k
views
Determinant of a rank $1$ update of a scalar matrix, or characteristic polynomial of a rank $1$ matrix
This question aims to create an "abstract duplicate" of numerous questions that ask about determinants of specific matrices (I may have missed a few):
Characteristic polynomial of a matrix ...
- 107k
164
votes
14
answers
41k
views
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 ...
anonymous
132
votes
7
answers
65k
views
Values of $\sum_{n=0}^\infty x^n$ and $\sum_{n=0}^N x^n$
Why does the following hold:
\begin{equation*}
\displaystyle \sum\limits_{n=0}^{\infty} 0.7^n=\frac{1}{1-0.7} = 10/3\quad ?
\end{equation*}
Can we generalize the above to
$\displaystyle \sum_{n=...
185
votes
9
answers
57k
views
How to define a bijection between $(0,1)$ and $(0,1]$?
How to define a bijection between $(0,1)$ and $(0,1]$?
Or any other open and closed intervals?
If the intervals are both open like $(-1,2)\text{ and }(-5,4)$ I do a cheap trick (don't know if that'...
- 2,073
364
votes
24
answers
44k
views
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:
$x>0$
$0^x=0^{x-0}=0^x/0^0$, so
$0^0=0^x/0^x=\,?$
Possible answers:
$0^0\cdot0^x=1\cdot0^0$, so $0^0=1$
$...
- 3,891
66
votes
16
answers
47k
views
Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using induction
How can I prove that
$$1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$$
for all $n \in \mathbb{N}$? I am looking for a proof using mathematical induction.
Thanks
842
votes
23
answers
102k
views
42
votes
8
answers
21k
views
Why $\gcd(b,qb+r)=\gcd(b,r),\,$ so $\,\gcd(b,a) = \gcd(b,a\bmod b)$
Given: $a = qb + r$. Then it holds that $\gcd(a,b)=\gcd(b,r)$. That doesn't sound logical to me. Why is this so?
Addendum by LePressentiment on 11/29/2013: (in the interest of http://meta.math....
- 1,611
340
votes
7
answers
45k
views
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/...
- 5,341
259
votes
32
answers
124k
views
Evaluating the integral $\int_0^\infty \frac{\sin x} x \,\mathrm dx = \frac \pi 2$?
A famous exercise which one encounters while doing Complex Analysis (Residue theory) is to prove that the given integral:
$$\int\limits_0^\infty \frac{\sin x} x \,\mathrm dx = \frac \pi 2$$
Well, can ...
user9413
130
votes
35
answers
287k
views
Proof $1+2+3+4+\cdots+n = \frac{n\times(n+1)}2$
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?
PS: This problem is known as "The sum of the first $n$ positive ...
- 1,467
106
votes
11
answers
8k
views
Closed form for $ \int_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$
I've been looking at
$$\int\limits_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$$
It seems that it always evaluates in terms of $\sin X$ and $\pi$, where $X$ is to be determined. For example:
$$\...
- 116k
208
votes
19
answers
127k
views
Proving $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx = \frac{\sqrt \pi}{2}$
How to prove
$$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2}$$
- 7,620
123
votes
3
answers
116k
views
Expected time to roll all 1 through 6 on a die
What is the average number of times it would it take to roll a fair 6-sided die and get all numbers on the die? The order in which the numbers appear does not matter.
I had this questions explained ...
- 1,395
11
votes
3
answers
2k
views
mod Distributive Law, factoring $\!\!\bmod\!\!:$ $\ ab\bmod ac = a(b\bmod c)$
I stumbled across this problem
Find $\,10^{\large 5^{102}}$ modulo $35$, i.e. the remainder left after it is divided by $35$
Beginning, we try to find a simplification for $10$ to get:
$$10 \equiv 3 ...
- 2,914
96
votes
15
answers
81k
views
How to prove that exponential grows faster than polynomial?
In other words, how to prove:
For all real constants $a$ and $b$ such that $a > 1$,
$$\lim_{n\to\infty}\frac{n^b}{a^n} = 0$$
I know the definition of limit but I feel that it's not enough to ...
- 6,575
62
votes
5
answers
48k
views
How to take the gradient of the quadratic form?
It's stated that the gradient of:
$$\frac{1}{2}x^TAx - b^Tx +c$$
is
$$\frac{1}{2}A^Tx + \frac{1}{2}Ax - b$$
How do you grind out this equation? Or specifically, how do you get from $x^TAx$ to $A^...
- 733
172
votes
17
answers
151k
views
How to prove Euler's formula: $e^{i\varphi}=\cos(\varphi) +i\sin(\varphi)$?
Could you provide a proof of Euler's formula: $e^{i\varphi}=\cos(\varphi) +i\sin(\varphi)$?
- 7,620
148
votes
2
answers
42k
views
Examples of 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.
- 31.3k
138
votes
32
answers
75k
views
Sum of First $n$ Squares Equals $\frac{n(n+1)(2n+1)}{6}$
I am just starting into calculus and I have a question about the following statement I encountered while learning about definite integrals:
$$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$$
I really ...
- 1,733
48
votes
3
answers
53k
views
Explain why $E(X) = \int_0^\infty (1-F_X (t)) \, dt$ for every nonnegative random variable $X$
Let $X$ be a non-negative random variable and $F_{X}$ the corresponding CDF. Show,
$$E(X) = \int_0^\infty (1-F_X (t)) \, dt$$
when $X$ has : a) a discrete distribution, b) a continuous ...
- 1,371
83
votes
15
answers
86k
views
Prove that $\lim \limits_{n \to \infty} \frac{x^n}{n!} = 0$, $x \in \Bbb R$.
Why is
$$\lim_{n \to \infty} \frac{2^n}{n!}=0\text{ ?}$$
Can we generalize it to any exponent $x \in \Bbb R$? This is to say, is
$$\lim_{n \to \infty} \frac{x^n}{n!}=0\text{ ?}$$
This is being ...
- 1,175
45
votes
8
answers
21k
views
Why is $a^n - b^n$ divisible by $a-b$?
I did some mathematical induction problems on divisibility
$9^n$ $-$ $2^n$ is divisible by 7.
$4^n$ $-$ $1$ is divisible by 3.
$9^n$ $-$ $4^n$ is divisible by 5.
Can these be generalized as
$a^n$ $-$...
- 557
251
votes
9
answers
29k
views
Evaluating $\lim\limits_{n\to\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$
I'm supposed to calculate:
$$\lim_{n\to\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!}$$
By using WolframAlpha, I might guess that the limit is $\frac{1}{2}$, which is a pretty interesting and nice ...
- 43.2k
124
votes
23
answers
24k
views
In classical logic, why is $(p\Rightarrow q)$ True if both $p$ and $q$ are False?
I am studying entailment in classical first-order logic.
The Truth Table we have been presented with for the statement $(p \Rightarrow q)\;$ (a.k.a. '$p$ implies $q$') is:
$$\begin{array}{|c|c|c|}
\...
- 1,359
29
votes
1
answer
4k
views
Discrete logarithm tables for the fields $\Bbb{F}_8$ and $\Bbb{F}_{16}$.
The smallest non-trivial finite field of characteristic two is
$$
\Bbb{F}_4=\{0,1,\beta,\beta+1=\beta^2\},
$$
where $\beta$ and $\beta+1$ are primitive cubic roots of unity, and zeros of the
...
17
votes
1
answer
9k
views
$\sum \cos$ when angles are in arithmetic progression [duplicate]
Possible Duplicate:
How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression?
Prove $$\cos(\alpha) + \cos(\alpha + \beta) + \cos(\alpha + 2\beta) + \dots + \cos[\...
- 179
171
votes
2
answers
96k
views
Discontinuous derivative. [duplicate]
Could someone give an example of a ‘very’ discontinuous derivative? I myself can only come up with examples where the derivative is discontinuous at only one point. I am assuming the function is real-...
- 1,827
153
votes
3
answers
23k
views
The square roots of different 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 ...
- 1,633
270
votes
1
answer
24k
views
How discontinuous can a derivative be?
There is a well-known result in elementary analysis due to Darboux which says if $f$ is a differentiable function then $f'$ satisfies the intermediate value property. To my knowledge, not many "...
- 8,133