Eric Thoma
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 Apr20 comment Characterization of measurable sets $E$ with $|E|_e<\infty$ Late to the party but: $\lvert E{\rvert}_e < \infty$ is needed for the other direction to show that when writing an open set $G \supset E$, $\lvert G-E\rvert < \epsilon$ as a countable union of nonoverlapping intervals, the tail of the union has small measure. This direction does not appear to need this, since the finite union of intervals is so nice and the other $N_1$ and $N_2$ are already small. Mar7 comment How could I prove whether the infinite series $n!/n^n$ converges or diverges? @JulianRachman "By this result, we can see that the series does not diverge, however it is still inconclusive that the series converges." This is not right at all. The theorem you used gives sufficient conditions for convergence, not necessary. And "converges xor diverges" is a tautology, so the statement itself does not make sense. (edit: it's fixed now, though only the ratio test is necessary, undid -1). Apr9 comment Prove that if b is coprime to 6 then $b^2 \equiv 1$ (mod 24) Since $b^2$ is a perfect square, we have that $b^2 = 1,4,9,16$ modulo $24$. All of these, besides $1$, require that $2 | b$ or $3 | b$. Mar9 comment Is this fraction even possible to put into partial fractions? @Wolff Think inverse trigonometric functions. Feb6 comment Real Analysis Question concerning existence of curve and derivative? Suppose $f(0) \ne 0$. Use continuity of $f$ to obtain a contradiction. Jan31 comment % of my equivalent human life hunting a fly for 5 minutes before I kill it, if its lived for 3 days and myself for 40 years House flies usually live for 15 to 30 days once hatched from pupal stage, though you may very well have been chasing a 3-day-old fly. Jan24 comment Does an element of a group to the 0th power equal the identity? In other words, the assignment of $x^0 = 1$ plays nice with $x^ax^b = x^{ab}$ for all $a$ and $b$, a desirable property. Jan19 comment Does this equation have positive integer solutions? In other words, combining any two Pythagorean triples gives a solution. Jan5 comment How do I show using math symbols “get quotient without remainder” Try $\lfloor 10/3 \rfloor=3$. This is the "floor" operation and gets rid of any decimal. Jan2 comment How find this integral Your bounds are for $t$, but you treated them as if they were for $\theta$. Notice that $\theta = \sin^{-1} t$, so you can easily switch back. Dec27 comment Don't be Greedier If a player removes one pebble when there is an odd number of pebbles remaining, that player wins. Dec27 comment Don't be Greedier What have you tried? Dec19 comment How to justify what solving method to use Every quadratic can be done by completing the square -- it is the equivalent of the quadratic formula I believe. Dec17 comment The intuitive understanding of $\sqrt{x}$ to model “inversely proportional / inverse square”? I am confused as to why my answer is being voted down. Is something I said incorrect or misleading? Dec16 comment Find derivative of $f(x)=(x+2)^{(x-1)}$ Others might be taking the natural logarithm of both sides and then using implicit differentiation (chain rule, really). As in: $f'(x) / f(x) = D_x ((x-1)\ln(x+2))$. Dec16 comment Show that the sum of the series I deleted my answer since my initial premise was faulty. Dec16 comment How many 6-letter words that have either exactly 2 vowels or 4 vowels are there? (all lower case) @mharris7190 If your follow-up is not already answered, I have answered above. For others, the follow-up was about viewing the problem from the perspective of putting distinguishable balls into distinguishable boxes without exclusion--a very good question. Dec16 comment How many 6-letter words that have either exactly 2 vowels or 4 vowels are there? (all lower case) Take it from the slot's perspective. It must choose one of the $5$ vowels. Since there are $2$ slots, there are $2$ factors of $5$, or $5^2$. The difference from ball-in-box problem is that multiple balls can go into each box, but slots cannot have multiple vowels. We can reduce it to a ball-in-box problem by saying the slots are the balls and the vowels are the boxes. Each slot picks a vowel to "go into". Notice that each vowel can "have" more than one slot, since there can be repeats--just like the ball-in-box problem! But now since the roles are reversed, the answer is $5^2$, not $2^5$. Dec16 comment How many ways to $22$ balls in $5$ boxes problem If the boxes are identical, the answer is the number of partitions of $22$. Dec16 comment Real life examples of commutative but non-associative operations This depicts $1 + 2 \ne 2 + 1$, a non-commutative system, not a non-associative one.