instruction stringlengths 12 30k |
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This is from the book Principles of Mathematical Analysis by Rudin, number 4 of chapter 7. It says consider
$$ f(x) = \sum\limits_{n=1}^{\infty}{ 1/(1+ n^2 x) } $$
The question asks:
(1) For what values of x does the series converge absolutely. We got that the series converges when x $\not=$ 0 & x $\not= -1/... |
Let $\mu (r)>2$ be the [irrationality measure](http://mathworld.wolfram.com/IrrationalityMeasure.html) of a transcendental number $r$, and consider the following set of points $P \in\mathbb{R}$:
$P=\{r\in \mathbb{R}: \mu(r)=Constant\}$
Is this set a fractal, and If so, then what is it's dimension?
|
A commutative ring $R$ with unity is Hermite if for all $x,y\in R$ there exists $t,u,v\in R$ such that $x=tu$, $y=tv$ and $(u,v)=(1)$. Is there a finite commutative ring with unity that is not Hermite?
This characterisation is taken from theorem 3 of:
Some Remarks About Elementary Divisor Rings, Leonard Gillman a... |
Is there a finite non-Hermite cru? |
Consider the following arithmetic function, $f(a,b) = (a+1, \lceil a- \frac{a}{b} \rceil , \lceil a- \frac{2a}{b} \rceil, \dots, \lceil \frac{a}{b} \rceil, 1)$, where all terms except the first and last are computed by the application of a ceiling function.
For any pair of positive integers $(a,b)$, the vector $f(a... |
Let $f(x)$ be a Lipschitz real-valued function defined on a closed interval $I$. The derivative $f '(x)$ exists a.e. since $f$ is absolutely continuous.
My question is: Is $f '(x)$ necessarily square integrable, i.e. in $L^2(I)$ ?
|
Is the derivative of a Lipschitz function square integrable? |
Consider the following arithmetic function, $f(a,b) = (a+1, \lceil a- \frac{a}{b} \rceil , \lceil a- \frac{2a}{b} \rceil, \dots, \lceil \frac{a}{b} \rceil, 1)$, where all terms except the first and last are computed by the application of a ceiling function.
For any pair of positive integers $(a,b)$, the vector $f(a... |
How does one show that any open interval in n-dimensional Euclidean space is connected?
This is homework, and I have been stuck on it for a few hours now. Seems like it should be easy, but I can't get a proof that I am comfortable with. |
Nate's example is a good one and actually solves the question as it is. But suppose you think that the Cromarty property, as Nate called it, is not good enough since 'almost all' spaces satisfy it. Then you try to substitute it for a stroger condition. You can require that every closed subset $Z \subseteq X$ is a count... |
Nate's example is a good one and actually solves the question as it is. But suppose you think that the Cromarty property, as Nate called it, is not good enough since 'almost all' spaces satisfy it. Then you try to substitute it for a stronger condition. You can require that every closed subset $Z \subseteq X$ is a coun... |
Nate's example is a good one and actually solves the question as it is. But suppose you think that the Cromarty property, as Nate called it, is not good enough since 'almost all' spaces satisfy it. Then you try to substitute it for a stronger condition. You can require that every closed subset $Z \subseteq X$ is a coun... |
Suppose $f$ is a real-valued function and $f(x) \geq 1$ for every real $x \in [0,1]$.
Why we can always find a unique positive integer $n$ such that $2^{n} \leq f(x) < 2^{n+1}$ ?
|
In answering a question I mentioned the Asteroids video game as the example -- at one time, the canonical example -- of a locally flat geometry that is globally different from the Euclidean plane. It might be out of date in 2010. This raises its own question:
are there other real-life examples of geometries formed... |
Im reading Number Theory I, by Kato, et al. The authors claim that for $x$, $y\in Q$ and $m$ a square free positive integer, if $2x$, $x^2-my^2 \in Z$, then for any odd prime $p$ $ord_p(m)+2ord_p(y) \geq 0$.
I get that $ord_p(my^2)=ord_p(m)+2ord_p(y)$ and $ord_p(x)\geq 0$. Im just not sure that I can say that $ord... |
Can you show that an interval is connected in $\mathbb{R}$? Then can you say (a,b) is connected to (c,b) is connected to (c,d)? |
In Andy's answer to the question "[What are fixed points of the Fourier Transform][1]" on Math Overflow, he shows that the Fourier Transform has eigenvalues {+1, +i, -1, -i} and that the projections of any function onto the corresponding four eigenspaces may be found through some simple linear algebra.
I would like ... |
How do I compute the eigenfunctions of the Fourier Transform? |
Now, suppose the transformation(in 2d) I am working with has two<br>
separate functions
for x and y.<br>
That is, the transformation for x is of the form<br>
x'=x/(x+y)<br>
and the transformation of y is<br>
y'=y/(x+y)<br>
Each is an LFT(The schwarzian derivatives are 0) but is<br>
the transform as a whole stil... |
is this a linear fractional transformation(LFT)? |
Nate's example is a good one and actually solves the question as it is. But suppose you think that the Cromarty property, as Nate called it, is not good enough since 'almost all' spaces satisfy it. Then you try to substitute it for a stronger condition. You can require that every closed subset $Z \subseteq X$ is a coun... |
I'm reading Number Theory I, by Kato, et al. The authors claim that for $x$, $y\in \mathbb{Q}$ and $m$ a square free positive integer, if $2x$, $x^2-my^2 \in \mathbb{Z}$, then for any odd prime $p$ $\mathrm{ord}_p(m)+2\mathrm{ord}_p(y) \geq 0$.
I get that $\mathrm{ord}_p(my^2)=\mathrm{ord}_p(m)+2\mathrm{ord}_p(y)$ ... |
I want to prove that for non-negative random variables with distribution F:
$$E(x^{n}) = \int_0^\infty n x^{n-1} P(\{X≥x\}) dx$$
Is the following proof correct?
$$R.H.S = \int_0^\infty n x^{n-1} P(\{X≥x\}) dx = \int_0^\infty n x^{n-1} (1-F(x)) dx$$
using integration by parts:
$$R.H.S = [x^{n}(1-F(x))]_0^\i... |
Moments and non-negative random variables? |
Here is how I have worked it out so far:
$log _2(x-4)+log(x+2)=4$
$log _2((x-4)(x+2)) = 4$
$(x-4)(x+2)=2^4$
$(x-4)(x+2)=16$
How do I proceed from here?
$x^2+2x-8 = 16$
$x^2+2x = 24$
$x(x+2) = 24$ Which I know is not the right answer
$x^2+2x-24 = 0$ Can't factor this
|
I want to prove that for non-negative random variables with distribution F:
$$E(x^{n}) = \int_0^\infty n x^{n-1} P(\{X≥x\}) dx$$
Is the following proof correct?
$$R.H.S = \int_0^\infty n x^{n-1} P(\{X≥x\}) dx = \int_0^\infty n x^{n-1} (1-F(x)) dx$$
using integration by parts:
$$R.H.S = [x^{n}(1-F(x))]_0^\i... |
I want to prove that for non-negative random variables with distribution F:
$$E(X^{n}) = \int_0^\infty n x^{n-1} P(\{X≥x\}) dx$$
Is the following proof correct?
$$R.H.S = \int_0^\infty n x^{n-1} P(\{X≥x\}) dx = \int_0^\infty n x^{n-1} (1-F(x)) dx$$
using integration by parts:
$$R.H.S = [x^{n}(1-F(x))]_0^\i... |
Here is how I have worked it out so far:
$\log _2(x-4)+\log(x+2)=4$
$\log _2((x-4)(x+2)) = 4$
$(x-4)(x+2)=2^4$
$(x-4)(x+2)=16$
How do I proceed from here?
$x^2+2x-8 = 16$
$x^2+2x = 24$
$x(x+2) = 24$ Which I know is not the right answer
$x^2+2x-24 = 0$ Can't factor this |
Perhaps you can apply the one found here: http://books.google.com/books?id=XqqNDQeLfAkC&pg=PA84
Snapshot:
![alt text][1]
[1]: https://i.stack.imgur.com/ySQ0q.png |
Do the [usual numerical phase retrieval algorithms][1] not work?
[1]: http://digitus.itk.ppke.hu/~matyi/optika/Phase_Diversity/AO82_PRComparison1.pdf |
x is the number of dropped connections, right? So $x=0$ for no dropped connections. |
You are dealt five cards from 52 card deck that is thoroughly shuffled. What is the probability that two out of five cards are seven?
Is it:
$${4C2 . 12C3 . (4C1)^3} / 52C5$$ |
Probability that two out of five cards are seven? |
In answering a question I mentioned the Asteroids video game as an example -- at one time, the canonical example -- of a locally flat geometry that is globally different from the Euclidean plane. It might be out of date in 2010. This raises its own question:
are there other real-life examples of geometries formed ... |
In answering a question I mentioned the Asteroids video game as an example -- at one time, the canonical example -- of a locally flat geometry that is globally different from the Euclidean plane. It might be out of date in 2010. This raises its own question:
are there other real-life examples of geometries formed ... |
I'd like to learn formal math. Preferably, though not necessarily, starting with predicate logic/first order logic rather than higher order logic. I am trying to find resources (papers, books etc.) for doing this, but I haven't found anything I really like.
There are lots of resources for predicate and first order ... |
The original Sonic the Hedgehog games (on the Sega Genesis and similar consoles) provide good examples. Many of the levels wrap around vertically, so that there is no bottom or top. They are equivalent to cylinders, I suppose.
Here are some examples from Sonic 3:
http://info.sonicretro.org/File:Ic1map.PNG
http... |
I'm trying to derive the integral form of the Bessel function by finding
the $k$th coefficient of the Laurent series expansion of the function
$f(z) =\exp [\lambda(z-\frac{1}{z})]$.
I managed to get it down to the form
$
J_k(\lambda) =\frac{1}{2\pi}\int_{0}^{2 \pi} e^{i(\lambda \sin\theta - k \theta)} d\theta
=... |
The original Sonic the Hedgehog games (on the Sega Genesis and similar consoles) provide good examples. Many of the levels wrap around vertically, so that there is no bottom or top. They are equivalent to cylinders, I suppose.
Here are some examples from Sonic 3:
http://info.sonicretro.org/File:Ic1map.PNG
http... |
Does the series $\sum\limits_{n=1}^{\infty}\frac{\sin(n-\sqrt{n^2+n})}{n}$ converge? |
On p.28 ([here][1]) Reid proves that the curve $y^2=x(x-1)(x-\lambda)$ for $\lambda\neq 0,1$ has no rational paramaterisation. At one point (on p.29), he has the equation $$r^2=ap(p-q)(p-\lambda q)$$ where $p$ and $q$ are polynomials over a field $k$, and $a$, $\lambda$ are constants in the field. He then says that "by... |
On p.28 ([here][1]) Reid proves that the curve $y^2=x(x-1)(x-\lambda)$ for $\lambda\neq 0,1$ has no rational paramaterisation. At one point (on p.29), he has the equation $$r^2=ap(p-q)(p-\lambda q)$$ where $p$ and $q$ are polynomials over a field $k$, and $a$, $\lambda$ are constants in the field. He then says that "by... |
If $A$ is a positive definite matrix, then the solution to the minimization problem $(1/2)x^TAx - b^Tx$ is given by $A^{-1}b$. I'm interested in the generalization of this to a Hilbert space. What conditions on $A$ should be required for this to be true in a Hilbert space? Also is there a reference that includes thi... |
A rectangle can be cut into two connected pieces of equal area so that they can be rearranged to get a new rectangle with different side lengths than the original one. Show that this can be done in countable many ways. Can this be done in uncountable many ways? |
I have a unit sphere, on the surface of which are defined spherical caps. I typically characterize the caps by the unit vector $n$ from the center of the sphere to the top of the cap, and the angle $\theta$.
My question is: given a pair of spherical caps, how can I determine whether they intersect? |
How to test whether spherical caps intersect? |
Today, I read these two new Logarithmic identity $\displaystyle$ $$a^{\log_a^m} = m $$ $$\log_{a^q}^{(m^p)} = \frac{p}{q} \log_a^m$$ Both of them seems new to me,so even after solving some problems (directly) based on thesm I haven't fully understood how they holds good,Could anybody show me how to prove them ?
|
Today, I read these two new Logarithmic identity $\displaystyle$ $$a^{\log_a m} = m $$ $$\log_{a^q}{m^p} = \frac{p}{q} \log_a m$$ Both of them seems new to me,so even after solving some problems (directly) based on thesm I haven't fully understood how they holds good,Could anybody show me how to prove them ?
|
The rule is that: **"Characteristic of a number($lt$ 1) with n zeros after decimal and before a significant digit = $\overline{( n + 1 )}$"**
When I am trying to verify with some random numbers like `Log[10, 0.0004356]` its giving me $-3.36091$ and `Log[10, 0.725]` gives me $-0.139662$. Giving $-3$ and $-0$ as cha... |
Suppose that K is a finite extension over $\mathbb{Q}$. Let $p$ be a prime in $\mathbb{Q}$, let $p\mathcal{O}_K=\mathfrak{P}_1\ldots\mathfrak{P}_n$ be its prime decomposition in $\mathcal{O}_K$, then is it true that any prime ideal in $\mathcal{O}_K$ above $p$ is a factor in the decomposition of $p$? |
The rule: **"Characteristic of a number($lt$ 1) with n zeros after decimal and before a significant digit = $\overline{( n + 1 )}$"**
When I am trying to verify with some random numbers like `Log[10, 0.0004356]` its giving me $-3.36091$ and `Log[10, 0.725]` gives me $-0.139662$. Giving $-3$ and $-0$ as characteris... |
Suppose that q is a rational prime, and K is a p-extension over Q where p is prime, then q ramifies in K if and only if q is p or q is congruent to 1 mod p. I don't know why this is true, can someone convince me of this? |
I was given a simple function, very simple. f(x) = bla bla. With a constant and an X. The point was to find for which values of a(the constant) the function is not one-to-one. The domain of the function is limited from zero and upwards. Here is what I put down on the paper:
As long as a is negative, it will not be ... |
Calculus, given function. Say when it's not one-to-one? (What did i miss) |
I was given a simple function, very simple. f(x) = bla bla. With a constant and an X. The point was to find for which values of a(the constant) the function is not one-to-one. The domain of the function is limited from zero and upwards. Here is what I put down on the paper:
As long as a is negative, it will not be ... |
Try to draw its graphic for $a=0$: maybe you'll recognise something familiar, won't you? Then, try to figure out what happens when you put some $a\neq 0$ (and you'll see that this doesn't change anything at all about injectivity).
Then you could try to think about this equation:
$$
(x_1 +a )^2 = (x_2 + a)^2 \ .
... |
We will only prove here that definition (1) implies definition (2) (this is the easy part).
Suppose that the condition in definition (1) holds. That is, $\mu_n ((a,b]) \to \mu ((a,b])$ for any $a,b \in D$, $a < b$, where $D$ is a dense subset of $R$. Define $\tilde D = \lbrace x \in R:\mu (\lbrace x \rbrace ) = 0 \r... |
Two numbers are equivalent if they have the same digits ex 2271 and 1227 are equivalent. How many equivalent classes are there between 1000 & 9999?
I'm not sure I fully understand how this works, is the answer to this question (9*10*10*10)/4!? First calculating the total numbers formed and divindg by 4! to get rid ... |
We will only prove here that definition (1) implies definition (2) (this is the easy part).
Suppose that the condition in definition (1) holds. That is, $\mu_n ((a,b]) \to \mu ((a,b])$ for any $a,b \in D$, $a < b$, where $D$ is a dense subset of $R$. Define $\tilde D = \lbrace x \in R:\mu (\lbrace x \rbrace ) = 0 \r... |
This question is about [Project Euler 113][1]:
> Working from left-to-right if no digit
> is exceeded by the digit to its left
> it is called an increasing number; for
> example, 134468.
>
> Similarly if no digit is exceeded by
> the digit to its right it is called a
> decreasing number; for example, 66420.... |
There are two things you are missing: you are missing the case in which $a\geq 0$; and you did not really justify your assertion when $a\lt 0$.
Technically, the only case you actually established was the case $a=-2$. That leaves quite a few values of $a$ untouched!
When one asserts that something happens "for all... |
First, you are trying to count increasing and decreasing numbers. They are related in that you only reverse the digits of one to get the other. The only difference is that increasing numbers cannot start with 0, but decreasing ones can end with 0. So let's just work on counting increasing ones.
How many one digit... |
One approach is to work by cases of the number of matching digits. If no digits match and none is zero, there are in fact 4!=24 permutations. So this accounts for 9*8*7*6/24 equivalence classes. If no digits match and one is zero, the zero cannot be the first digit, so there are only 3*3*2*1=18 permutations and 1*9*... |
The rule: **"Characteristic of a number($\lt$ 1) with n zeros after decimal and before a significant digit = $\overline{( n + 1 )}$"**
When I am trying to verify with some random numbers like `Log[10, 0.0004356]` its giving me $-3.36091$ and `Log[10, 0.725]` gives me $-0.139662$. Giving $-3$ and $-0$ as characteri... |
The [rule][1]: **"Characteristic of a number($\lt$ 1) with n zeros after decimal and before a significant digit = $\overline{( n + 1 )}$"**
When I am trying to verify with some random numbers like `Log[10, 0.0004356]` its giving me $-3.36091$ and `Log[10, 0.725]` gives me $-0.139662$. Giving $-3$ and $-0$ as chara... |
First, you are trying to count increasing and decreasing numbers. They are related in that you only reverse the digits of one to get the other. The only difference is that increasing numbers cannot start with 0, but decreasing ones can end with 0. So let's just work on counting increasing ones.
How many one digit... |
Original equation:
$g(x) = \frac{(x+1)(x^2+2)(x^3+3)}{\sqrt{x^4+3}}$
If I take ln on both sides, and than differentiate I get this:
$\frac{1}{g(x)} = \frac{1}{x+1}+\frac{1}{x^2+2}+\frac{1}{x^3+3}-\frac{1}{2}* \frac{1}{x^4+3} $
Is this good so far? I don't know what to do about the left side? |
Calculus, differentiation, but first taking ln on both sides? |
What is an example of a **continuous** $2\pi$ (or 1) periodic function whose Fourier series converges pointwise but not uniformly? (Such function cannot be of Holder class, or absolutely continuous.) I am thinking that maybe the Cantor function? |
The [rule][1]: **"Characteristic of a number($\lt$ 1) with n zeros after decimal and before a significant digit = $\overline{( n + 1 )}$"** This rule is mentioned in my module,Goggle lead me to this [link][1].
When I am trying to verify with some random numbers like `Log[10, 0.0004356]` its giving me $-3.36091$ and... |
Under the isomorphism $L^2(S^1)\to\ell^2(\mathbb Z),\quad e^{2\pi i n t}\mapsto e_n$, smooth functions on the circle are mapped to rapidly decaying sequences (see [wikipedia](http://en.wikipedia.org/wiki/Fourier_series#Properties)).
Is the converse also true? That is, does every rapidly decaying sequence in $\ell^2(... |
Does rapid decay of Fourier coefficients imply smoothness? |
There are two things you are missing: you are missing the case in which $a\geq 0$; and you did not really justify your assertion when $a\lt 0$.
Technically, the only case you actually established was the case $a=-2$. That leaves quite a few values of $a$ untouched!
When one asserts that something happens "for all... |
Under the isomorphism of Hilbert spaces $L^2(S^1)\to\ell^2(\mathbb Z),\quad e^{2\pi i n t}\mapsto e_n$, smooth functions on the circle are mapped to rapidly decaying sequences (see [wikipedia](http://en.wikipedia.org/wiki/Fourier_series#Properties)).
Is the converse also true? That is, does every rapidly decaying se... |
This question is about [Project Euler 113][1]:
> Working from left-to-right if no digit
> is exceeded by the digit to its left
> it is called an increasing number; for
> example, 134468.
>
> Similarly if no digit is exceeded by
> the digit to its right it is called a
> decreasing number; for example, 66420.... |
I am given $n$ points in 2D.(Each of say approximately equal weight). I want to partition it into $m$ clusters ($m$ can be anything and it is input by the user) in such a way that the center of mass of each cluster is "far" from center of mass of all other clusters. What is a good heuristic approach (it should also be ... |
In "A Short Course on Spectral Theory", page 10, William Arveson asserts that the "ax+b group", ie. the group generated by all dilations and translations of the real line, is isomorpic to the group of all (real) 2x2 matrices of the form
a b
0 1/a
a>0, b real.
It is very easy to check that the ax+... |
> If set is closed under some operation,
> what does that tell us about stricture
> of groupoid?
It's a question from one of the previous math exams which was shown to me by one of my friends.
Unfortunately, I can't find a good answer to it. As far as I know, the answer is nothing, because of the definition of... |
If set is closed under some operation, what does that tell us about stricture of groupoid? |
> If set is closed under some operation,
> what does that tell us the about structure
> of the groupoid?
It's a question from one of the previous math exams which was shown to me by one of my friends.
Unfortunately, I can't find a good answer to it. As far as I know, the answer is nothing, because of the defin... |
If set is closed under some operation, what does that tell us about the structure of the groupoid? |
What is an example of a **continuous**, or even better, differentiable, $2\pi$ (or 1) periodic function whose Fourier series converges pointwise but not uniformly? (Such function cannot be of Holder class, or absolutely continuous.) I am thinking that maybe the Cantor function? |
This actually triggered me in my mind from [here][1]. After some playing around I notice that the relation $a^{\log_b x} = x^{\log_b a}$ is true for any valid value of $a,b$ and $x$. I am very inquisitive to see how this holds ?
[1]: http://math.stackexchange.com/questions/10850/solving-an-equation |
How $a^{\log_b x} = x^{\log_b a}$ ? |
I am quite bafelled now, I am not getting by what it is written that $log_10 \tan 40^\circ \cdot log_10 \tan 41^\circ \cdot log_10 \tan 42^\circ \cdotlog_10 \tan 43^\circ \cdots log_10 \tan 50^\circ = log_10 \tan 40^\circ + log_10 \tan 41^\circ + log_10 \tan 42^\circ + log_10 \tan 43^\circ \cdots log_10 \tan 50^\circ... |
Can we write this ? |
I am quite bafelled now, I am not getting by what it is written that
\begin{align*}
&\log_{10} \tan 40^\circ \cdot \log_{10} \tan 41^\circ \cdot \log_{10} \tan 42^\circ \cdot\log_{10} \tan 43^\circ \cdots \log_{10} \tan 50^\circ \\\
&= \log_{10} \tan 40^\circ + \log_{10} \tan 41^\circ + \log_{10} \tan 42^\circ ... |
In algebra, we learn that if a function $ f(x) $ has a one-to-one mapping, then we can find the inverse function $ f^{-1}(x) $. The method that I have seen taught is the "horizontal line test": if any horizontal line touches the graph of the function more than once, then it must not be one-to-one. This method is not ex... |
I am quite a baffled now, I am not getting by what it is written that
\begin{align*}
&\log_{10} \tan 40^\circ \cdot \log_{10} \tan 41^\circ \cdot \log_{10} \tan 42^\circ \cdot\log_{10} \tan 43^\circ \cdots \log_{10} \tan 50^\circ \\\
&= \log_{10} \tan 40^\circ + \log_{10} \tan 41^\circ + \log_{10} \tan 42^\circ... |
I am quite a baffled now, I am not getting by *how* it can be written that :
\begin{align*}
&\log_{10} \tan 40^\circ \cdot \log_{10} \tan 41^\circ \cdot \log_{10} \tan 42^\circ \cdot\log_{10} \tan 43^\circ \cdots \log_{10} \tan 50^\circ \\\
&= \log_{10} \tan 40^\circ + \log_{10} \tan 41^\circ + \log_{10} \tan 42^... |
Find equation of curve fit programmatically in Matlab? |
do you have an example of a non separable locally compact connected metric space?
Thank you |
What is an example of a **continuous**, or even better, differentiable, $2\pi$ (or 1) periodic function whose Fourier series converges pointwise but not uniformly? (Such function cannot be of Holder class, or absolutely continuous.) |
First, let me answer your question; but please keep reading, because there's lots more to say.
Yes: there is an analytic way to see if a function is one-to-one. For this, you need the "analytic definition" of being one-to-one. The definition is:
$$f\text{ is one-to-one if and only if for all }a,b\text{ if }a\neq ... |
First, let me answer your question; but please keep reading, because there's lots more to say.
Yes: there is an analytic way to see if a function is one-to-one. For this, you need the "analytic definition" of being one-to-one. The definition is:
$$f\text{ is one-to-one if and only if for all }a,b\text{ if }a\neq ... |
First, let me answer your question; but please keep reading, because there's lots more to say.
Yes: there is an analytic way to see if a function is one-to-one. For this, you need the "analytic definition" of being one-to-one. The definition is:
$$f\text{ is one-to-one if and only if for all }a,b\text{ if }a\neq ... |
Intuition is developed by doing examples and becoming familiar with these things, so in that respect you are heading in the right direction.
As far as "guessing the limit function", that's not what we want to do! We don't want to guess, we want to *figure it out*.
Certainly if there is going to be uniform converg... |
Intuition is developed by doing examples and becoming familiar with these things, so in that respect you are heading in the right direction.
As far as "guessing the limit function", that's not what we want to do! We don't want to guess, we want to *figure it out*.
Certainly if there is going to be uniform converg... |
How to prove the following theorem?
Theorem:
If the binomial numbers C(n,p) and C(n,q) are equal, with p different from q, then p+q = n.
(n, p and q are natural numbers)
|
How to prove the following theorem?
Theorem:
If the binomial numbers $C(n,p)$ and $C(n,q)$ are equal, with $p$ different from $q$, then $p+q = n$.
($n$, $p$ and $q$ are natural numbers)
|
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