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6h
answered Definite integral with cube roots of trig functions
21h
comment Is there $n$ such that $n,n^2,n^3$ start with the same digit ($\neq 1)$
Here's one way to work it out. You want the fractional parts of $\ln n,\,\ln n^2,\,\ln n^3$ to be approximately equal, so $n$ should be near a power of $10$. We're now allowed to go just over, so go just under instead. If $n=10^k-1$ then $n^3\approx 10^{3k}\left(1-3\times 10^{-3k}\right)$, so $k=1$ won't help but $k=2$ will.
1d
comment how can we define closed set or open set for a set of matrices?
@jaggu Let $S$ denote the desired set of $\Delta<0$ matrices. Andre's comment noted an inner product on $\mathbb{R}^{2\times 2}$, which we can use to define a distance metric. Let $B\left(A,\,r\right)$ denote the radius-$r$ open ball in $S$ of centre $A$. Since $\Delta$ is continuous in all entries of $A$, for sufficiently small $r>0$ we have $B\left(A,\,r\right)\subseteq S$.
1d
comment how can we define closed set or open set for a set of matrices?
@jaggu You're considering the set of $A$ satisfying $\Delta\left(A\right)<0$. Can you see why such a set would contain a neighbourhood of each element?
Feb
11
revised Can an infinite sum of irrational numbers be rational?
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Feb
10
answered How do I determine if A is diagonalizable
Feb
10
comment What are some surprising appearances of $e$?
To make $e$ out of this, take determinants with $t\text{tr} A = 1$.
Feb
10
answered number of real solution of $\sin x+2\sin 2x = 3+3\sin 3x\;\forall x\in \left[0,\pi\right]$
Feb
10
answered Can an infinite sum of irrational numbers be rational?
Feb
9
answered Find the derivative of $y=\frac{\tan(x)}{1+\tan(x)}$
Feb
9
answered Propositional-Calculus/ Set Theory Proof using Identities
Feb
8
answered Contradiction! Any Symbol for?
Feb
7
answered Prove that $\frac{7}{12}<\ln 2<\frac{5}{6}$ using real analysis
Feb
6
answered Show the given series is a solution of $y''-xy'-y=0$
Feb
6
comment solution of differential equation $\left(\frac{dy}{dx}\right)^2-x\frac{dy}{dx}+y=0$
@NoChance That's the special case $a=0$.
Feb
6
revised solution of differential equation $\left(\frac{dy}{dx}\right)^2-x\frac{dy}{dx}+y=0$
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Feb
6
answered solution of differential equation $\left(\frac{dy}{dx}\right)^2-x\frac{dy}{dx}+y=0$
Feb
5
answered Solve $x^2 = 2^x$.
Feb
5
answered Catherine is now twice as old as Jason but 6 years ago she was 5 times as old as he was. How old is Catherine now?
Feb
5
revised How to get this result of integral?
deleted 104 characters in body