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.
Homework Statement
A is an mxn matrix and C is a 1xm matrix. Prove that CA=Sum of (C sub j)*(A sub j) from j=1 to m.
Where A sub j is the jth row of A
Sorry for the messy problem statement, I couldn't figure out the summation notation on here.
Homework Equations
The Attempt at a...
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.
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