[eqn]\frac{8}{\pi}\int_0^\infty \cos \left(2x\right)\prod_{n=1}^\infty\c

>>8527751

"It's less than 10^(-41) from 1, yet it's not 1."

well, no shit. In stead of asking us how, try proving it yourself and if you can't prove it,try showing us what you did or talk to a professor. 3/10 if troll

>>8527751
How the fuck would you even integrate infinite products?

>>8527780
Calc II, the post

>>8527780

I don't know the proof of this off the type of my head, but I have an idea of the direction to take. in terms of integrating infinite products, it usually doesn't make sense to do it directly. There's probably an identity some where i'm not aware of or a transform. The very precise nature of the product of this integral is a bit odd. I'm going to try and figure out a proof of it now and in a few hours (3-4) i'll come back with a proof or the working of a proof. Keep the threadalive because I would like to see other people's posts

>>8527780
>>8527815
http://math.stackexchange.com/questions/151997/how-to-calculate-this-infinite-integral-of-infinite-product-of-cosine

if, for [math]a,b,c \in {\R}[/math] a number [math]c[/math] exists so that [math]a<c<b[/math], then [math]a \neq b[/math] . Otherwise [math]a=b[/math].

This should be obvious. I can easily think of a number between that decimal and 1. For example, [math]0.999999999999999999999999999999999999999999822\dots[/math], thus your big decimal number is not equal to 1.


You can also use this (not very rigorous) definition to show that [math]0.9\bar=1[/math], because there exists no number in between them where [math]0.9\bar<c<1[/math].

>>8527826
>unrelated tangent from the first week of real analysis

K

>>8527830
Pedants gonna pedant

>>8527851
>fucks up basic latex
>makes an irrelevant post
>literally uses grade school math and has the gall to say "this should be obvious"
>claims that criticism of this retardation is pedantic

Shiggity

>>8527826

everything you said is unrelated or wrong. 2/10

How did you calculate this?

File: pf.jpg (35KB, 382x271px) Image search: [Google]

35KB, 382x271px

check this out. This should a sufficient proof for your statement

>>8527876
>numerical analysis
>sufficient/convincing proof

My God, sci is really going to shit

File: pf2.png (51KB, 408x934px) Image search: [Google]

51KB, 408x934px

>>8527876

a more complete analysis

>>8527826
I dont think anyone was wondering why 0.99997 or some shit is not 1

>>8527879

fuck off, it's the best I could find. I'd be more than happy to read a more rigorous, complete proof that you provide

>>8527876
>>8527881

Here's the source:

http://www.ams.org/notices/201110/rtx111001410p.pdf

scroll down to page9 and you;ll find it.

https://en.wikipedia.org/wiki/Borwein_integral

>>8527924

good link. to expand on that, here are some articles published by respected mathematicians and engineers:

http://www.thebigquestions.com/borweinintegrals.pdf

http://schmid-werren.ch/hanspeter/publications/2014elemath.pdf


I mentioned earlier that I would try coming up with a proof of op's statement but after browsing this thread, I can say that aint happening

>>8527951
>respected
>engineers

Also
>I mentioned earlier that I would try coming up with a proof of op's statement but after browsing this thread, I can say that aint happening

Pussy

>>8527780
Integrate to some unknown value t and then find the limit as t approaches infinity

>>8527879
>several decimal digits of disagreement

>>8527751

In one sense, the more incredible thing about doing calculus with the elementary functions is that in specific, nice conditions, they give "nice", compact answers not just /at all/, but on a very regular basis, things like 0, 1, 2, 3, 2e, 3π/8, and so on.

Forget everything you know about math for a moment, in order to run a thought experiment. When confronted with some strange, complex formula like the LHS of OP's equation, one that (for our purposes of discussion) you are told is supposed to represent some number, or maybe infinity, or something like that, one would not immediatey expect just by looking at the formula itself that it is in fact equal to some "simple" number like 1, as is in fact the case here assuming OP's formulae are right.

Moreover, staying speculative and artificially naive a moment (when of course we know the following to be true), it is easy to imagine that there an infinity of strange, complex formulae like the OP's which are really really close to some such number, but are in fact not equal to that number.

>>8527780

Recall that (in one treatment) whenever infinity is written, this does not actually invoke the strange notion of infinity-as-such, but instead is a shorthand for a /limit/ of some kind. One is thus obliged to recall the limit laws, and how they may behave with respect to the operators entailed.

From this, one ought also to verify that one actually understands what the integrand is, and what it (literally) looks like, in terms of a graph, if such is feasible. You had better be able to evaluate f(0), f(1) etc, for example, and identify any weird or pathological points, if you propose to integrate.

Not having worked the problem, I don't claim that the above observations lead directly to a proof, or solution. But they are clearly prerequisites, things that you ought to be able to do if you propose to prove such-and-such, which is a more involved thing, and will necessarily entail some of the above at some point.

>>8528299

By the second paragraph I of course meant to indicate that OP's such-and-such is of course /not/ supposed to be equal to one, as he wrote. My clauses became confused around this point and can be taken to mean that I thought that OP's thing /is/ exactly equal to one, which is false according to what the OP has written.

>>8527881
[eqn] \frac{8}{\pi}\sum_{m=0}^\infty\int_0^\infty \cos \left(2(2m+1)x\right)\prod_{n=1}^\infty\cos\left(\frac{x}{n}\right)dx = 1 [/eqn]

Thats a nice identity. Also explains why >>8527751 isnt one nicely.