alex.jordan
Reputation
28,359
381/400 score
 1d comment Use the definition of convergence of a sequence to show $\lim \frac {2n^2}{n^3+3}= 0$ @mathreadler It can be argued that when $a_n$ is understood to be a sequence (as opposed to a function of real $n$ or some other object), that $\lim a_n$ is unambiguously asking for the limit as $n\to\infty$. But it would come down to convention. 1d comment Use the definition of convergence of a sequence to show $\lim \frac {2n^2}{n^3+3}= 0$ If you have access to the Squeeze Theorem, the argument can be fairly short. Do you? 2d comment Prove that 10101…10101 is NOT a prime. @MXYMXY If the number of $1$s is even, then $101$ divides such a number. If the number of $1$s is $2k+1$, ThomasAndrews shows that you have $\overbrace{11\cdots1}^{2k+1}\cdot\overbrace{9090\cdots9091}^{2k}$. 2d comment Find to how many digits the value $\frac{355}{113}$ is an accurate approximation of $3.1415929204$. In general, the following is valid. Since you have to memorize six digits for the continued fraction convergent $\frac{355}{113}$, then you will have something with six digits of decimal accuracy. And that is exactly true for this particular convergent. Feb 6 comment Why does $\sum\limits_{n=0}^{+\infty} z^n=\frac{1}{1-z}?$ https://en.wikipedia.org/wiki/Geometric_series Feb 6 answered A number theory contest problem Feb 5 revised Are there singular matrices such that if we change any entry it will be non-singular? added 754 characters in body Feb 5 answered Are there singular matrices such that if we change any entry it will be non-singular? Feb 5 comment Are there singular matrices such that if we change any entry it will be non-singular? When you say "if one changes any single entry" do you mean that in the sense that the entry could be changed to anything and make the determinant nonzero? Or that for any entry, there would be some number you could change it to that would make the determinant nonzero? Feb 5 comment Are there singular matrices such that if we change any entry it will be non-singular? This isn't enough. You could have a $3\times3$ matrix with the first two columns identical and an independent third column. Then the rank is $n-1$, but altering the right column entries does not make the matrix nonsingular. Feb 3 comment Suppose that $f(0)=f(2\pi)$. Show that there exists an x such that $f(x)=f(x+\pi)$. What goes up must come down. Feb 3 comment Find all functions so that $f\left(\frac{x}{f(y)}\right) = \frac{x}{f(x\sqrt{y})}$ Your question says "find a function $f$ so that..." which means that if you can find even one function (like $x^{1/2}$ which results from this investigation) then you have replied to the question. As posted, your question says nothing about finding "the only" function with the given property, so there is no burden to prove uniqueness (if that's even true). Feb 3 answered Find all functions so that $f\left(\frac{x}{f(y)}\right) = \frac{x}{f(x\sqrt{y})}$ Feb 2 comment What is the difference between independent and mutually exclusive events? They are, in a sense, completely opposite features. If $A$ and $B$ are independent, knowledge that $A$ occurred does not change the probabilities that $B$ may have occurred. Where as if $A$ and $B$ are disjoint, knowledge that $A$ occurred completely changes the probabilities that $B$ may have occurred by collapsing them to $0$. Feb 2 comment Determine the general solutions of $x'''-5x''+8x'-4x=0 , t∈R$ Was this exercise in a textbook? If so, I'm sure there are examples very similar to this if you read the section. Feb 2 awarded Revival Feb 2 comment Why is $\sum_{n=1}^\infty {1\over{n^{1+ {1\over \ln n}}}}$ divergent? Note that your series is undefined at the $n=0$ term, unless you interpret $1/\ln(0)$ as $0$. Feb 2 answered Jordan form of the matrix $\left(\begin{smallmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \end{smallmatrix}\right)$ Feb 1 revised What's the meaning of $C^1(R)$? added 1 character in body Feb 1 answered What's the meaning of $C^1(R)$?