Recent content by CalTech>MIT

  1. C

    Is the Metric Space (X,d) Separable and Compact?

    How exactly would you prove this using the original equation?
  2. C

    Is the Sequence {a_n} Convergent Given Its Recurrence Relation?

    I'm not very good at this stuff, so guys tell me if I am wrong somewhere, but here it goes: 2an \leq an-1 + an+1 Through rearrangement: an - an+1 \leq an-1 - an This means that the difference between successive terms is decreasing. Since an is decreasing, the differences must decrease to...
  3. C

    Solve Series Proof Homework: 0<b<1 Convergence & Limit

    Homework Statement Let 0<b<1, show that \sum^{n}_{r=1} (1/rb - \frac{n1-b}{1-b}) converges as n goes to infinity and denote the limit by \beta = \beta(b). Also, show that \sum^{infinity}_{n=1} \frac{(-1)n-1}{nb} + \beta(21-b - 1) = 0Homework Equations The Attempt at a Solution Absolutely...
  4. C

    Does the Convergence of \(\Sigma \frac{f(x)}{x}\) Depend on \(\Sigma f(x) = 0\)?

    Homework Statement Let f:Z\rightarrowR be periodic such that f(x+a) = f(x) for some fixed a\geq1. Prove that \Sigma ^{infinity}_{x=1} \frac{f(x)}{x} converges if and only if \Sigma ^{a}_{x=1} f(x) = 0. Homework Equations n/a The Attempt at a Solution Ok, so I have a general...
  5. C

    Is the Sequence {a_n} Convergent Given Its Recurrence Relation?

    First, Caltech>MIT lol. Also, I believe you'd have to assume that the sequence an is bounded to solve this problem.
  6. C

    Is the Metric Space (X,d) Separable and Compact?

    I believe it'd be: (x1,1, x2,2, x3,3, ...)?
  7. C

    Is the Metric Space (X,d) Separable and Compact?

    Homework Statement X={x | xn E R | 0\leq x \leq 1} d(x,y)= \Sigman=1infinity |xn - yn|*2-j Show: 1. (X,d) is a metric space 2. (X,d) is separable 3. (X,d) is compactHomework Equations n/aThe Attempt at a Solution Here we go. number 1. Show that d(x,y)=d(y,x): \Sigman=1infinity |xn - yn|*2-j =...