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Thread: Verifying/Disproving Calculus Solution

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    Verifying/Disproving Calculus Solution

    Hello all,

    Strange way to introduce myself, but I didn't want to create a banal intro thread.

    I recently took a calc test and there is one specific problem that seems to have multiple answers depending on which university presents it. Out of curiosity I read into this problem in-depth and I think I have arrived at a solution that is mathematically sound (and computationally verifiable for the first thirty five billion iterations), however I don't have anyone to share it with that cares/understands the nature of it.

    Wolfram alpha, alongside every other program I know has failed to produce the correct result, or I'm missing something. I was hoping that someone would either prove or disprove my solution. I think it is an interesting problem, however I know math isn't everyone's cup of tea so I apologize in advance for boring some (if not most) of you.


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    Hmm. I haven't done maths for flipping ages.

    I think you're "right", but maybe a few parts of your proof are a bit off.

    As I said, I haven't done maths for flipping ages, so I could be missing something.

    When you introduce the Cauchy Condensation Test, what are you actually trying to show? Because (as Wikipedia just tells me), this is an iff to prove that a series if convergent. You just used it, showed that the "condensed" series converges, and therefore the original also converges (seemingly this is sufficient to prove the original question, and leave it at that). But right before you introduced the Cauchy Condensation Test, you seemed to be saying you were about to show that the original series is less than the n^1.0001 thing.

    Actually, at this point, I believe there's an error. You've taken the "condensed" (Cauchy Condensation Test transformed series) and calculated its value to be the irrational number 3.6095... Then you're saying the "original" series is also that same number.

    [edit: Actually, my mistake. I should read the second paragraph on wikipedia. "Moreover, if they converge, the sum of the condensed series is no more than twice as large as the sum of the original."]

    Perhaps I'm just out of the game for a while. But, I think it would also generally help to just add comments like "doing this because...". Used to really piss me off when maths professors would be as absolutely minimal/uncommented as possible, maybe because "it's obvious". E.g. when you do the "proportional to" thing.

    Also, towards the end, where you do the "let's make n really big" thing. I may have missed the line of your reasoning, however, I can't quite see what the formal mathematical process of proof is that you're using here? Other than "it's obviously correct", which, it seems to be, but often you need to do a bit better than that.

  3. #3
    TJ TeresaJ's Avatar
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    I'm out of practice when it comes to calculus, but I just wanted to pop in to say welcome! Maybe you could also make a welcome thread.

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    Meae Musae Servus Hephaestus's Avatar
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    You are using Wolfram Alpha wrong.

    You also miss the low hanging fruit. I'd have tossed the coefficient from the condensation and observed n^(-3) < n^(-2) therefore it converges therefore so does the original sum.

    I know you were shooting for more, but graded papers (or worse, tests) are a high risk place to get creative. Do that in office hours where it might count for something other than irritating the grader looking for the obvious solutions to problems chosen for having solutions obvious to those competent in the tested subject matter. No one likes having their jerb forcibly desimplified by some rando seeking validation for being smrt.

    Your proof is also very very messy IMSO. Read some Alfred Whitehead or Burden & Faire's "Numerical Analysis" and you'll see what good, lucid proofs look like. It'll give you something to strive toward. I highly recommend Numerical Analysis just for the opening chapter's approach to precise definition of a limit. Stewart isn't bad, but it's a more difficult read for a novice. B&F have a more approachable tack that is still elegant and makes Stewart's approach much more tractable.

    Whitehead is just an excellent example of how to write clearly and plainly about maths.
    People think they understand their own mortality, even when that understanding has just changed.

    --Meditations on Uncertainty Vol ξ(x)

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    I appreciate the feedback everyone. While I was "right" (after presenting to a few PhDs) the mathematics I used were outside of the scope of the course, resulting in me contriving a banal solution to the originally proposed problem.

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    Meae Musae Servus Hephaestus's Avatar
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    Quote Originally Posted by zeta View Post
    I appreciate the feedback everyone. While I was "right" (after presenting to a few PhDs) the mathematics I used were outside of the scope of the course, resulting in me contriving a banal solution to the originally proposed problem.
    Yeah. That's one of the trickier things about proof classes. I had an absurdly hard time in a class on graph theory because I walked in already knowing too much from previous classes--it's really frustrating to have to prove something without access to a basic fact you have primed and loaded to springboard from. Sometimes it's bit like having only one leg in a butt-kicking contest that requires you to keep one foot on the floor at all times.

    Ultimately I found that most of the time, the problems were all supposed to have fairly simple solutions, so if you start feeling clever, you're probably on the wrong track--but that doesn't mean it's a bad train of thought to follow, it just might not be what you want to submit. If you aren't trying to solve most of the problems multiple ways, you aren't trying.

    Also worth noting, "in vino veritas" doesn't hold up when writing proofs. Or at least, most of my late-night wine-fueled veritas didn't survive proof-reading* in the morning.


    *
    People think they understand their own mortality, even when that understanding has just changed.

    --Meditations on Uncertainty Vol ξ(x)

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