Recent content by kolley

Because it's discontinuous at all integers.

If it's equal to an integer then it would not be differentiable.

Yes sorry that was a typo, should be (0,1). So would I set k=[1/x], which would make f(x)=x2*k which would imply that f'(x)=2xk Is this what you mean?

Homework Statement k(x)=x2*[1/x] for 0<x≤1 k(x)=0 for x=0 Find where k(x) is differentiable and find the derivative Homework Equations The Attempt at a Solution I know that it is differentiable for all ℝ\Z on (0,1], but I am unsure how to find the derivative for this problem.

how does L'Hopital's rule get incorporated into a formal proof for a sequence converging. I don't understand

Homework Statement prove (n+1)/(3n+1) converges to 1/3 Homework Equations The Attempt at a Solution I have been trying to figure this out for a while. I started out (n+1)/3n+1)-1/3=(2n+2)/(9n-3) now I don't know how to proceed with the proof. What do I set N equal to? And...

Homework Statement x^2+y^2=z^2 Homework Equations The Attempt at a Solution assume to the contrary that two odd numbers squared can be perfect squares. Then, x=2j+1 y=2k+1 (2j+1)^2 +(2k+1)^2=z^2 4j^2 +4j+1+4k^2+4k+1 =4j^2+4k^2+4j+4k+2=z^2 =2[2(j^2+K^2+j+k)+1)]=2s the...

I guess the factorial is what is throwing me off, I don't know how to use a chain of inequalities that will lead me to something that I can directly compare to 2^k+1 because I don't know how to take the factorial into account or get rid of it.

sorry, i meant for n=4

Homework Statement prove n!>2^n for all n>=4 Homework Equations The Attempt at a Solution I showed it was true for n=1. assume k!>2^k for all k>=4 then show it for k+1. (k+1)!>=2^(k+1) =k!*(k+1)>=2*2^k I don't know where to go from here.

I thought the way to do it was by contradiction. But I'm confused as to how to produce a generalized irrational number, and then like you say, get one smaller than that.

Homework Statement Prove that there is no smallest positive irrational number Homework Equations The Attempt at a Solution I have no idea how to do this, please help walk me through it.

Sorry, I left out part of mine. I had 49/36 +1/(k+1)^2 <= 2-1/(k+1) since 1+1/4+1/9 is equal to 49/36, is this correct or am I still on the wrong track?

Homework Statement prove that 1+1/4+1/9+...+1/n^2< or = 2-1/n for every positive integer n Homework Equations The Attempt at a Solution proved it was correct for n=1, then replaced the n with k, changed it to k+1 to get: 1/(k+1)^2 < or = 2-1/(k+1) don't know how to proceed

I have no idea how to go about this, except that we are supposed to use proof by contradiction to show that the square root of 3 is an irrational number. Any help or tips is appreciated. Thanks.