Recent content by jrsweet

  1. J

    X is an accumulation point show there is subsequence that converges to x

    Can we say the sequence is {x-1/k2} for integers>0 ? I'm still a little confused. I know that the interval (x-1/k, x+1/k) will be contained in a neighborhood of x, and thus has infinitely many points. I'm not sure why we are using 1/k though. I guess we are in essence using the definition of...
  2. J

    X is an accumulation point show there is subsequence that converges to x

    Homework Statement Suppose x is an accumulation point of {an: n is a member of integers}. Show there is a subsequence of (an) that converges to x. The Attempt at a Solution I'm a little stuck on this one. I know that since x is an accumulation point then every neighborhood around x...
  3. J

    Prove x belongs to the set or is an accumulation point.

    So, I might have got it... I tried it by contradiction. Assume x is not a member of S and assume it is not an accumulation point of S. If there is a neighborhood (x - epsilon, x + epsilon) containing x that does not have a point of S, then (x - epsilon) is an upper bound of S that's less than...
  4. J

    Prove x belongs to the set or is an accumulation point.

    Homework Statement Let S be a nonempty set of real numbers bounded from above and let x=supS. Prove x either belongs to the set or is an accumulation point of S. Homework Equations x is an accumulation point of S iff each neighborhood of x contains a member of S different from x. That...
  5. J

    Prove 2^(-n) converges

    I was confused how to find it that way. I tried this: 1/2N < e 1/e < 2N ln(1/e) < Nln2 ln(1/e)/ln2 < N does that work?
  6. J

    Prove 2^(-n) converges

    Chose e>0. Let N be any positive integer greater than 1/e. Then, for n>=N we have |1/(2n)-0| < |1/n|= (1/n) <= (1/N) < (1/(1/e)) = e Thus, the sequence converges to 0?? Does that look right?
  7. J

    Prove 2^(-n) converges

    Homework Statement Using the definition of convergence to prove that the sequence {2^(-n)} converges Homework Equations The Attempt at a Solution So, I just don't think I am thinking straight or something. Here is what I got so far: Chose e>0. Let N be any positive integer...
  8. J

    Between any two distinct real numbers there is a rational number

    So, we chose q>1/(y-x)? So then, 0 < p-(x/(y-x)) < 1 0 < p < (y/(y-x)) I'm still lost on how we choose p then.
  9. J

    Between any two distinct real numbers there is a rational number

    Homework Statement Let x and y be real numbers with x<y and write an inequality involving a rational number p/q capturing what we need to prove. Multiply everything in your inequality by q, then explain why this means you want q to be large enough so that q(y-x)>1 . Explain how you...
  10. J

    Piecewise continuous - step function

    Homework Statement Consider the following initial value problem: y''+4y = 9t, 0<=t<2 ............0, t>=2 Find the equation you get by taking the Laplace transform of the differential equation and solve for Y(s) Homework Equations The Attempt at a Solution I am just having trouble with the...
  11. J

    Laplace transform initial value problem

    Homework Statement Use the Laplace transform to solve the following initial value problem: x' = 7 x + 5 y, y'= -2 x + e5t, x(0)=0, y(0)=0 Find the expressions you obtain by taking the Laplace transform of both differential equations and solving for Y(s) and X(s) Homework Equations...
  12. J

    Laplace transformation t^(3/2)

    Homework Statement Just a quick question concerning a Laplace transformation... Find the Laplace transform of the following function: f(t)=10t3/2-e(-7t) Homework Equations The Attempt at a Solution I wasn't sure what to do with the t3/2 so I just followed the formula for t1/2...
  13. J

    Multiple eigenvalue solutions

    Homework Statement Solve the system. dx/dt=[1 -4; 4 -7]*x with x(0)=[3; 2] Homework Equations The Attempt at a Solution I am apparently not getting this at all. Can someone walk me through it? I konw I have to first find the eigenvalues and eigenvectors: (1-λ)(-7-λ)+16=0 λ2+6λ+9=0...
  14. J

    Eigenvalue Method

    Homework Statement solve the system dx/dt = [12 -6; 6 -3] with the initial value x(0) = [12; 9] Homework Equations The Attempt at a Solution I know I need to find the Eigenvalues but then I get a little confused from there. (λ-3)(λ+3)=0 λ=3, -3
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