# JavaMan

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I am not here with any regularity.

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 1d comment Find the limit $\lim_{n \to \infty}\left(\frac{a^{1/n}+b^{1/n}+c^{1/n}}{3}\right)^n$I have edited your answer as it was quite startling to see such huge fonts. If you want to see the $\TeX$ I used, you can right click on the math and select "Show Math As -> TeX Commands" 1d revised Find the limit $\lim_{n \to \infty}\left(\frac{a^{1/n}+b^{1/n}+c^{1/n}}{3}\right)^n$deleted huge and large fonts, aligned equations, change e^x wiith exp(x), changed log to \log 1d comment Number Theory $8 \mid (a^2-b^2)$ for $a$ and $b$ both oddEven easier: $a \in \{1,3,5,7\} \implies a^2 \equiv 1 \pmod{8}$. May18 revised Proof using trigonometry that circle circumference is $2 \pi R$fixed typos May18 comment Proof using trigonometry that circle circumference is $2 \pi R$I should also mention that Michael Hardy mentioned that the circumference of a circle and its length are often confused, and I admit that I use the two terms interchangeably here. What I give up in precision, I make up for in getting my point across. May18 answered Proof using trigonometry that circle circumference is $2 \pi R$ May18 comment Proving an inequality: $|1-e^{i\theta}|\le|\theta|$+1: IIRC, the OP's problem appears in "Berkeley Problems in Mathematics" and your hint appears as their solution. May18 comment A binary quadratic form: $nx^2-y^2=2$ May17 awarded Constituent May17 comment Proof using trigonometry that circle circumference is $2 \pi R$What is $\pi$ if not the ratio of a circle's circumference to the length of its diameter? May17 comment Proof f(x) is continuous given $x$ rational and irrational.Hint: Show that if $a \neq \frac{1}{2}$, then $\displaystyle \lim_{x \to a} f(x)$ does not exist. May17 comment Proof using trigonometry that circle circumference is $2 \pi R$How do you define $\pi$? May17 comment Using the hypothesis $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}$ to prove something else+1, though I would add the words "Without loss of generality" before $a = -b$. The reason is that it may well not be the case that $a = -b$, but we know at least one of $a = -b, b = -c$, or $a = -c$ occurs, and all cases are dealt with similarly. May16 comment How to prove that $1/n!$ is less than $1/n^2$?Correction: the induction works for $n \geq 4$ as $3^2 \not< 3!$. May16 reviewed Approve suggested edit on Test of convergence of $\int_{-\infty}^{\infty} \dfrac{x^6+6}{x^8+8}dx$ May16 answered Legendre symbol proof May16 revised Solving systems of equations using matricesadded 589 characters in body May16 comment Solving systems of equations using matricesI'm not sure I understand what you mean. Gauss-Jordan gives you a method for reducing all augmented matrices regardless of how messy the matrix. In other words, if you always follow the steps of Gauss-Jordan, you will eventually arrive at a matrix which is in reduced row echelon form. May16 comment Piecewise defined integration$f_n(x) = \frac{n}{2} \cdot 1_{\left[ - \frac{1}{n}, \frac{1}{n} \right]}(x)$, where $1_A(x)$ is the indicator function. May16 answered Solving systems of equations using matrices