This is a thread for everyone who likes to torture themselves, and more importantly others, with various math problems.

I just had my Real Analysis final today that I was terrified for and surprisingly did okay on.

These were two problems that at first glance seemed impossible but after thinking about them they were very straight forward and are the only reasoning I’m going to pass the final.

I don’t know if anyone here would be interested in this type of stuff, but I thought I’d post it anyways.

Without further ado!

If some sequence {an} converges to 0.

Where {an} is the same sequence in the problem without the fancy subscript business and the arrow implies convergence.
I have trouble getting text to line up correct in latex so I wrote as much as I could outside of it. Still a noob at it.

This was directed towards the strange individuals who do math for fun. I don’t expect many to be, but I can hope!

Just trying to see who the other mathochists on the forum are…if any at all.

I would much rather talk about/solve a physics problem than do math like this. This was honestly the worst class I’ve ever taken and it hurt my math heart bad

Theoretical stuff and proofs just aren’t my cup of tea.

I hear you there. I just want my math to get the job done. I don’t really care to solve for proofs and what not. I like using it to conceptualize relationships between different things.

The implication there is that 0.9 repeating is exactly equal to 1, which you’ll find to be perfectly consistent

I’ll take a crack at the problems! I haven’t taken real analysis, but I can cheat my way through with basic calculus. Forgive my lack of rigor, I do physics, not mathematics. There’s lots of glossing over the finer details in that subject.

You have magnificently beautiful handwriting. I wish my own was close to that neat

Number 1 leaves a little bit more to be desired. Your lower bound looks perfect but your upper bound could be a bit more precise. Can you think of an f(x) that might be simpler to integrate that also works as an upper bound?

Number 2 looks good! I’d question your upperbound again, but I see that you have the right idea when you state it’s bounded above. I think you accidentally wrote your lowerbound twice

I misread your upperbound.

I apologize for not having the skills to do this in LateX nor the handwriting to do this on paper.

I started number 2 by first bounding (-1)^n above and below. which would give you -1 < (-1)^n < 1 and then you just multiply through by {an}. It’s easier this way since you don’t have to bother with the positive/negative cases and you still get to use the squeeze theorem at the end of the day.

Ahh I remember doing something similar with a car on an offramp back in general physics.

Say I were to solve your problem with a Lagrangian, would I use a holonomic constraint R - r = 0 ? Where r is the radius of the biker.

…1/x^4 would have been super easy now that you mention it. Silly me.

You can try to apply a Lagrangian to my problem if you really want to, but it’s my intuition that you’d have a really nasty time trying to extract a friction coefficient from it.

I’ll try to come up with an entertaining Lagrangian problem tomorrow!

Its a shame how many people don’t like maths, not that many people realize that maths can be fun
Does anyone here watch any of numberphiles vidios? I found some of them to be quite fun.

When I found out that 0.9999… was the same as 1 I had great fun confusing some of the less mathematically inclined of my friends with it. I even showed them some proofs for it but some still didn’t get it.

Believe it or not, there are some people who actively and aggressively deny this. It’s not just about “not getting it”. They’re active denialists. I’ve been trying to figure out why people would go out of their way to fight this idea for the better part of 15 years now.

Yeah, I understand why people don’t understand. Infinities and infinitesimals really throw people off. I don’t understand why some people get so aggressive about it!

To understand it, think back to your reaction to @NavyFish’s bringing up LaViolette’s Subquantum Kinetics. That represented an assault on your view of the world. I suspect that the more acceptance it gained, the more uncomfortable you would be with it.

Once we accept something as true, we start to build on it and base other ideas, concepts, memories, plans and such on that something. It becomes a foundation concept. When that foundation is shaken, it scares people. Different people have different reactions. Some get aggressive.

Being told that 0.9999… equals 1.0 is ludicrous on the face of it. I understand the simple mathematical proof of it, but that doesn’t make it sound any more reasonable.

Yeah, except that didn’t present “an assault on [my] world view”, it presented a model I hadn’t heard of, and that I had a few basic questions about. Navy addressed the questions to my satisfaction, and the theory is now on my radar. I didn’t shout it down, I didn’t refuse to believe it in the face of proof, and I certainly don’t think it’s ludicrous.

I have no strong feelings on it either way, and I’m certainly not chasing down proponents of it to argue how it simply cannot be true because I don’t understand it. And it’s an idea that really is contrary to my current understanding of the world, and would have implications in my life if it turned out to be true. 0.999… = 1 doesn’t challenge anyone’s anything.

I suspect that your world view is unassailable because you aren’t the sort to rely on foundation concepts. They are simply working principles for you, subject to revision as new proof comes to light. Other people - and I’d assert, most people - are given to examining something only so long as they resolve it to the point where it’s not of concern to them, then they rely on them as foundation concepts.

I brought up that example because it is the closest that you likely get to being in the shoes of someone who gets aggressive over being told something like 0.999… equals 1.0.