Because I didn't note any easy way to show that I can place the limit inside the integral, I used that way I showed you @robjohn $\lim_{a\to 0} \int_0^{\infty} \dfrac{1-e^{-a y}}{y}\arctan(1/y)$.
That's why I also asked @DanielFischer. Maybe @TedShifrin can explain that swapping ...
Maybe I miss an obvious way (out for some sleep).
@robjohn at any rate, I'm glad I have that elementary way. No need to explain a swapping in this not-comfortable case.
I'll assume you are working with sequences or functions on the natural numbers. If "constant increase of 2" means $f(n)+2=f(n+1)$, then it would be an arithmetic progression like $3,5,7,9,11,\dots$. If "constant increase of 2" means the rate increases by 2, then it would be something more like $1,3,7,13,21,\dots$
even so, I still feel like its offset a bit. If as you say $f(n)=1+2T(n)$, then $f(n+1)-f(n)=2n+2$ instead of $2n$. It seems rather to be $f(n)=1+2T(n-1)$
In general, if you start adding at $1$ and stop adding at $n$, it is what we call the $n^{th}$ triangle number, often denoted as $T(n)$, and will equal $\frac{n(n+1)}{2}$
Lets see if it is infact offset at all., $f(0)$ should be 1., so I'm thinking it will be $f(n)=1+2T(n)$ afterall. $f(0)=1+2T(0) = 1+2\cdot0 = 1$., and then $f(1)=1+2T(1) = 1+2\cdot 1 = 3$., $f(2)=1+2T(2) = 1+2\cdot 3 = 1+6=7$, $f(3)=1+2\cdot 6 = 1+12 = 13\dots$
$f(n)=n^2$ goes $1,4,9,16,25,36,49$, it has the correct rate increase but the increase starts too fast.
$f(n)=n^2-n$ goes $0,2,6,12,20,30,42$ has the correct rate increase and initial increase. but it starts too small.
$f(n)=n^2-n+1$ fits perfectly.
@Chantry @Ilya_Gazman the difference between the answer posted and mine is a difference of $2n$. In fact, his answer goes $1,1,3,7,13,\dots$ where $f(0)=1$ and $f(1)=1$. It was my understanding that $1$ corresponded to the $0^{th}$ term and $3$ corresponded to the $1^{st}$ term.
The way you phrased the question in your questionpost does not mention that the sequence starts with the $0^{th}$ term, and the answerer assumed that it was starting from the $1^{st}$ term
@Karlo I didn't ask if you tried, I asked what you don't understand. the left side of the equation is a union, and the wikipedia article links to a page about unions. is that what you were having trouble with?
It seems that A is always the number of things and the difference between the left side which is uniting sets and the right thing on the right side which is intersecting sets are opposite?the second one shows how many elements have this property and the other how many don't
@Lepidopterist The overwhelming scientific consensus on the subject is that vaccine usage is good. Any articles floating around the internet claiming that "vaccines cause autism" should generally be ignored as there isn't even a correlation between the two in the first place.
@JMoravitz i don't care about media consensus, i care about good research. I am suspicious of bullying in science, and I never said anything about autism
@Lepidopterist unless you're going to do all of the relevant science yourself or read and evaluate all of the relevant literature yourself, you are relying on scientific consensus, for all the science topics out there.
i have many reasons to look into medical statistics by myself. i simply do not trust the math and methodology of many of their studies. look at the nutrition field, for instance. i'm afraid in this day in age my trust is limited when it comes to personal health
The incidence of apparent autism is lower than the incidence of health problems based on sickness. The autism thing is based off of ONE falsified(for monetary gain) study. Autism isn't as bad as your kids dying and/or killing other kids who have immune issues. The list goes on @Lepid
What in particular are you concerned about with regards to vaccines? Their effectiveness in preventing disease? Their potential ineffectiveness against certain diseases? Their supposed link to autism? Their potential link to other side-effects?
@Lepidopterist Well you have to have done little to no research before coming here and talking about this, so I will just drop out of this conversation
i'm concerned that they are a multi-billion dollar operation and there seems to be little transparency. i am concerned about the increase in the incidence of many neurological illnesses in my lifetime.
anyway, i didn't draw any conclusions. strange that i'm getting a backlash.
@Lepidopterist it is not strange you are getting backlash; the majority of people using your talking points are trolling or rationalizing a dangerous medical position
@Lepidopterist "I don't trust scientists; it's all about the money; they're hiding things; what about the side effects, etc." are the ones that don't include autism. balance and proportion are two different things: you can balance pros with cons, or you can look at how big they are relative to each other.
you talk about the overall field, but then don't seem aware that vaccines are one of the most fundamental crowning achievements of all of medical science - not some obscure topic with inconsistent results in the literature that the media has been making spinny buzz about ever since measles and polio.
it takes some serious muster-passing to become textbook and not just somewhere on the record in the literature
@anon you must be well aware of the difference between the pure vaccine and the delivery methods used. and the difference between "one dose" vaccines and multi-dose vaccines which require more than just the dose to keep them fresh. i know enough about the biology and epidemiological results to know that many vaccines have an effect
there is nothing inconsistent with vaccines being powerful tools with their having dangers, or with particular ways of delivering the vaccine having dangers
My current stance is skepticism. I have only read epidemiological studies so far (a tricky, unreliable methodology usually). It's suggestive, but I am looking to see how vaccines are tested for side-effects. Some have famously given adverse reactions to people. I have experience with known drugs that go to market that are later recalled because of new data. I want to be careful
i'm not a biologist, I study math. Anyway, I do not want to be seen as trolling anyone. I'm genuinely interested. We can drop the subject
According to this YouTube video, viruses are actually man-made nano-technology.
The author clearly understands biology and the traditional understandings of viruses and the immune system, yet he still claims that viruses are not natural, but products of human technology?
I however think that p...
It's interesting, though, how suspicious common people have become of medicine. I think there was a period when most people trusted doctors and medical research
If you tolerate skeptics you will have to tolerate some cranks. Heretics drive scientific revolutions, though. Dyson has a good essay on that
If there's anything more insufferable than a crank, it's the "I f***ing love science" crowd that parrots whatever they think intelligent people should be saying, without thinking for themselves
A lot of beliefs high-IQ people adopt are meant to signal their intelligence to other high-IQ people
Quick question: Why is the Legendre symbol multiplicative? Specifically the case when $a, b$ are not both squares modulo $p$? Is there an intuitive way to understand this without resorting to Euler's Criterion?
An editor, that makes you type math, faster than you can write it. Some kind of wysiswyg hybrid, so that you see what you are actually writing. Something that exports to LaTeX.
I often talk with parenthesis to clarify things(like this) in my standard chat text, and I looked up in this chat and found many other people also do this here, so to see if it was a math thing I checked my uni math page, and many of the students were also doing it there. I have not seen this on any other uni page(non math)
"There was last year when [name redacted] took it, but before that, there wasn't usually any number theory. [name redacted]'s an awesome teacher, but he often sets very challenging, time-consuming assignments - in many ways, this is a good thing, because you'll learn a lot through doing that, and what's the point of doing a course if it's not going to challenge you? (When I had him for this course in 2013, he set weekly assignments that each took about 5 hours to complete; there was a lot of mixed feedback out of that, so I'm not sure if he's going to…
@robjohn I'm just trying to understand what you said since I know no such theorem. I don't say it's wrong, I'm investigating (and vaguely remember a counterexample, but just vaguely).
@robjohn The approach by generating functions is horrible. As in many other cases, the nicest approach is by using series, avoiding as much as possible the use of the integrals.
@robjohn I know! This happens to me too while working on some nice questions. I mean I start with a question and at the end of the solution I realize I have another problems ready to tackle. :-)
@robjohn Did you see that $$\int_0^1\left(\pi x\cot(\pi x)+\frac2{1-x^2}\right)x\,\mathrm{d}x=\int_0^1\left(\pi \cot(\pi x)+\frac2{1-x^2}\right)x\,\mathrm{d}x$$? :D
It is given the following algebraic curve:
$$ZY^2=X^3+3XZ$$
I want to find the group of rational points of finite order $E(\mathbb{Q})_{\text{torsion}}$ and to determine if $E(\mathbb{Q})$ is finite or infinite.
I thought the corresponding curve in affine coordinates is: $Y^2=X^3+3X$ with $a=3...
Ich will mich für die mündliche Prüfung von das Große Deutsche Sprachdiplom vorbereiten.
Am Anfang gibt es den persönlichen Teil in dem man sich vorstellen muss und von sich was erzählen. Was könnte man zum Beispiel sagen?
I want to prepare myself for the oral examination of Großes Deutsche...
@robjohn Trying to explain all (like swapping integrals order) in my book will probably turn my book into a mess at least for 2 reasons: 1). some proofs become pretty large and complex while doing that 2). the reader might be scared and abandon many nice proofs because of these details that might turn a $2$ pages proof or less into a $4-5$ pages proof.
It took me more than 4 pages to explain all. @robjohn I simply cannot go this way.
@robjohn I hope I won't annoy anyone saying that, but one needs to be a good professional to get that completely and perfectly done, explaining all the steps. Some readers might have problems following that way.
$f(n)=n^2$ goes $1,4,9,16,25,36,49$, it has the correct rate increase but the increase starts too fast.
$f(n)=n^2-n$ goes $0,2,6,12,20,30,42$ has the correct rate increase and initial increase. but it starts too small.
$f(n)=n^2-n+1$ fits perfectly.
According to this YouTube video, viruses are actually man-made nano-technology.
The author clearly understands biology and the traditional understandings of viruses and the immune system, yet he still claims that viruses are not natural, but products of human technology?
I however think that p...
It is given the following algebraic curve:
$$ZY^2=X^3+3XZ$$
I want to find the group of rational points of finite order $E(\mathbb{Q})_{\text{torsion}}$ and to determine if $E(\mathbb{Q})$ is finite or infinite.
I thought the corresponding curve in affine coordinates is: $Y^2=X^3+3X$ with $a=3...
Ich will mich für die mündliche Prüfung von das Große Deutsche Sprachdiplom vorbereiten.
Am Anfang gibt es den persönlichen Teil in dem man sich vorstellen muss und von sich was erzählen. Was könnte man zum Beispiel sagen?
I want to prepare myself for the oral examination of Großes Deutsche...
