Tanner Swett
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 Apr 26 comment Shouldn't all alternating series diverge by the diverge test? To be precise, the limit of the sequence $b_n$ is 3/4, but the sequence $a_n$ does not have a limit. You're right that when it comes to determining whether or not a sequence converges to 0, sign isn't an issue. Apr 26 answered Shouldn't all alternating series diverge by the diverge test? Apr 19 answered How do you create an alternating series with the sign being the same twice in a row? Apr 10 answered I don't understand what a “free group” is! Apr 3 comment Is there such a thing as backwards sigma? But that's why I said "if it appears twenty times in your summand". In your two examples, the subtraction only appears once. If it appeared twenty times, it would start to get cumbersome. Mar 26 comment Is there such a thing as backwards sigma? Doing subtraction like this isn't necessary the best notation. If $(11 - i)$ appears twenty times in your summand, it's probably best to think of an alternative. Feb 10 comment What is the flaw of this proof (largest integer)? That's correct, Chris. Aug 31 comment Relation between XOR and Symmetric difference True. Restating myself more precisely: there's a one-to-one correspondence between the subsets of a set and the predicates on that set, but in a set theory such as ZFC, not every predicate defines a set. Aug 27 awarded Good Answer Jul 19 awarded Yearling Jul 9 answered W. Mückenheim claims a severe inconsistency of transfinite set theory; true? Jul 9 comment In a vector space in finite dimension, all vectors which are not colinear, are orthogonal. True or false? The word "orthogonal" is not defined for vector spaces, so the question uses undefined terminology, and must be modified before it makes sense. Your argument is valid if you replace "all vectors that are not collinear are orthogonal" with "for every pair of vectors that are not collinear, there is an inner product for which they are orthogonal". Jul 6 answered Properties that are true for finite sets but are (non-trivially) false for infinite sets Jul 2 comment Why are conformal mappings necessarily 1 to 1? Well, both the unit disk and the upper half-plane have the same cardinality, so there's no problem there. The upper half-plane has infinite area, while the unit disk has finite area; but biholomorphic functions don't have to preserve area, so that's not a problem, either. If you're familiar with an order-preserving bijection between the open interval $(0,1)$ and the entire real line, this biholomorphic function acts much the same way. Jun 11 comment Conflicting limit answers using calculator and wolfram alpha I don't think this answer is quite correct. A calculator that uses floating point numbers is unlikely to round a very small number to zero. My guess is that $\sin 0.0000000001$ and $\tan 0.0000000001$ are so close together that the calculator rounded the two numbers to exactly the same number, meaning that it subtracted two identical numbers, getting zero. May 21 answered Floor function to the base 2 May 11 awarded Popular Question May 5 revised Number of Distinct Axiomatic Systems Describe construction of theories with models of arbitrary finite size May 5 answered Simple Adding and Subtracting algorithm to get a current amount May 5 comment Number of Distinct Axiomatic Systems It can be written out explicitly by an algorithm. Here's a first-order theory where every model has exactly four elements: "a ≠ b. a ≠ c. a ≠ d. b ≠ c. b ≠ d. c ≠ d. For all e, e = a or e = b or e = c or e = d." You can do the same for any number n: think of n constants, then write down axioms asserting that no two of the constants are equal to each other, and then write down one more axiom asserting that everything equals one of the constants.