You can show its converging with the alternating series test i linked.

The 1/(5n+1) diverges because it can be bounded by a series with terms like 1/5n which is like 1/5* the harmonic that diverges. You can show it diverges by using some integral argument if you are familiar with the process because the integral 1/x goes like log(x) or you can do that to the original 1/(5x+1) anyway. Do you know what i mean? Like here

http://en.wikipedia.org/wiki/Integra...or_convergence

http://en.wikipedia.org/wiki/Series_...vergence_tests



You just showed it diverges so what do you mean by conditionally convergent now, it isnt.

So you apply as you said next below the same alternating test for (-1)^n/ln(n).

Yes I realized you can show it by integral test.

And sorry, what I meant was how to check whether (-1)^n/ln n is cond. convergent. You can use alt. test series yes?

Assume that the limit as x ---> c does not exist, because it diverges to opposite infinities from either side (e.g., 1/x as x -----> 0).

Question: Does this limit fail to exist because x is not approaching the same finite value from either side, or does it not exist because it diverges to infinity?

Because it diverges to infinity.

If it were |1/x| it would be going to the same infinity but the limit still wouldn't exist.

Doesn't it go to +∞ from the right and -∞ from the left?

The absolute value flips the left side above the x-axis, giving you the Atari logo.

Oops. Didn't see the abs value signs.

This is not a hw question, im jw.

I somehow ended up watching this video:

"Euler Solves The Basel Problem"
http://www.youtube.com/watch?v=NmSBnOaAjjQ

And im wondering, I'm 2 weeks from my final in calculus 1 (derivatives mostly) but we are now doing integrals. Should I be able to use google and understand this video? Or should I be expected after finishing calc 2 to use google to understand it? Or should i not be able to understand it unless I've done grad school in math. I guess I mean comprehend more than understand. B/c right now I do not comprehend wtf he did

I'm finishing up Calc 2 right now, and I must disappoint you, you most likely won't get to that, in much depth at least.
You, of course, spend a lot of time talking about infinite series like that. You do touch on power series, and Taylor etc. by the end of the term. You learn basics, though, and for more rigid explanations you'll have to go deeper into mathematics.

But by the end of the next term when you take calc 2, even though you wouldn't be able (probably) to do that on your own, it'll be very clear what's going on in the video.

I hope it helps.

Thanks, and yes, that helped a lot xD

Aside from the Taylor series which you haven't gotten to yet, that explanation requires nothing but high school algebra which you should understand now. That's the simplest explanation there is because it's not rigorous, i.e., Euler assumed that he could do things to infinite polynomials which he could do to finite polynomials, and that was not justified until 100 years later. Making that explanation rigorous requires more advanced math, and there are different proofs which also require more advanced math. Here is Euler's simple method written down concisely so you don't have to listen to 20 minutes of rambling. You know how to factor a difference of squares, and you know how to factor a polynomial, and that's all he's using aside from the Taylor series. I don't know whether you'll get to Taylor series in your next class or not. You probably will since they usually stick it in there somewhere early, unlike when I was in school when sequences and series were an entire course taken after multivariable calculus.

Sequences and series was my favorite part of both calculus courses fwiw.

It would be cool to each challenge ourselves to come up with an original proof of Sum(1/n^2)=Pi^2/6 using basic math or geometry and then of course any advanced (say calculus plus complex etc) math method we like for fun.

That might be a bit too challenging for most of us. In particular since the problem has been around for so long and many "simple" proofs have been found.

Maybe pick a simpler problem.

If one needs to pick a problem that is easy, where exactly is the victory when you find it? A victory only matters if its something of significant effort or ingenuity or an opportunity to learn new things or it serves a decent purpose etc you get/know the idea.

I agree that people like to pick easy problems to solve in order to declare easier a victory but i dont care about winning. Students at school often grow up under the illusion all problems are solvable somehow lol. Its a very profound cultural naivete that the educational system does very little early on to attack. I care about the adventure even if that proves i am not good enough.

Also i think i trust most of us are ethical enough to not try to find all elementary methods in books or online and claim them as their own with some tricky modification etc but originally find themselves there if so. In fact i suppose a truly original method even if it shares a lot with an old one can be probably revealed as original if it appears to be a little bit more complex or not efficient enough etc. You see what i mean? Very often the first success is tortured, sometimes you can tell if its plagiarism or an honest coincidence that is unavoidable given that most elementary methods and arsenal are common to all people who love math anyway and its unavoidable to think similarly often.


Besides the real challenge is one to yourself and needs no witnesses. If you then find it elsewhere then so be it, it will still count in the only eyes that matter.

I suggested it only as some fun activity if you ever find yourself anywhere and want to kill some time (eg traveling or waiting somewhere) or take a break from other realistic tasks.

Furthermore one can take a challenge in many different ways. You could even define it as finding a way to estimate it that is very fast for example.

You mean like when President Garfield found an original proof of the Pythagorean theorem? And how long had that problem been around? It was a good one too. He put 2 right triangles together to make a trapezoid, then found the area of the trapezoid.

http://www.curiouser.co.uk/facts/garfield_proof.htm

He might have been our smartest president. Too bad someone had to shoot him 4 months into office.

I have t₁ and t_n, respectively the largest and smallest t values in a sequence of t values that is n long

I need to calculate t_i i=2,...,n-1 so that t_i i=1,2,...,n-1,n are equidistant.

All I found so far is some geometry method didn't seem to be what I'm looking for.

It's for c++ code...

There are n-1 intervals, so first find the length of each interval by dividing the total difference between t_n and t₁ by n-1. Then add the correct multiple of that interval onto t₁ to get t_i.

MdZ, this challenge is going to be only a huge time-eater

After a couple of hours of thought I realised that the formula commonly used (in conjuction with L'Hopital's rule) to solve the Basel problem - the fractional representation of the cotangent that was proven via Fourier series in my course of analysis - can be proved simpler by observing that pi*cot(pi*x)-1/x and the fractional series for it satisfy the same functional equation f(x) = 0.5*f(x/2) - 0.5*f((1-x)/2) + 1/(1-x) on [0,1/2] and then showing that this equation has a unique continuous solution up to a constant difference - but then I googled and found out that this proof is called the Herglotz trick. Did you know it? I for one didn't, but it's quite intuitive, and I dealt with a similar kind of equations (scaling equations) in wavelet analysis.

Bottom line: most simple proofs of known facts have already been discovered as we're not the most intelligent mathematicians in history; so it makes little sense to try to reinvent the wheel.

The one using the Fourier series for x and Parseval's rule is very simple.

http://en.wikipedia.org/wiki/Basel_p...Fourier_series

I agree that it's the simplest way, and that's how I'd solve the problem at an exam... I was just trying to make up a more 'original' argument, but that's just futile, all simple solutions to this problem are likely already known.

Half the complaints dealt with plumber A, who does 40% of the plumbing jobs in the town.

How should I be interpreting this? Is it P(A|Complaint) = 0,5 or P(Complaint|A) = 0,5 or P(A and Complaint) = 0,5. Seems like it makes sense both ways...

P(A|Complaint) = 0.5.

P(Complaint|A) = P(Complaint & A)/P(A) = P(A|Complaint)*P(Complaint)/P(A)

= 0.5*P(Complaint)/0.4

We don't know P(Complaint). Say it's 10%. Then P(Complaint|A) = 12.5%. So he generates 12.5% complaints on 40% of the jobs or 5% of the complaints which is half the complaints as advertised, and that's P(Complaint & A).

Why do those guys use minus signs for equal signs. That's extremely confusing.

I see equality signs in the .pdf, viewing it with the Chrome plugin; the horizontal bars are however so close that a low resolution reader / monitor might show them as merged into thick minus signs.