Recent content by rainwyz0706

Could you please be a bit more specific about your second line? The k here means the k-th derivatives. The power of (x-1)(x+1) is a fix n, and I don't think I'm supposed to do an induction on that. f^(r)(x) has to be a pretty messy function, is there a clear way to take derivative out of that?

I tried to prove by induction on r. But I'm not sure how to express the k-th derivative of f(x). r=0 or r=n are special cases, they clearly holds. My problem is how to generalize it.

Let f (x) = (x^2 − 1)^n . Prove (by induction on r) that for r = 0, 1, 2, · · · , n, f^ (r) (x)(the r-th derivative of f(x)) is a polynomial whose value is 0 at no fewer than r distinct points of (−1, 1). I'm thinking about expanding f(x) as the sum of the (n+1) terms, then it's easier to...

Thanks, I got it!

Thanks for your reply. I think clearly y=x is not uniformly convergent, so I guess kn(x) isn't either? About the second one, I tried to work with the epsilon-delta definition, but the result seemed still depend on n. Could you please be a bit more specific how you would do it?

1.kn (x) = 0 for x ≤ n x − n, x ≥ n, Is kn(x) uniformly convergent on R? I can show that it is uniformly convergent on any closed bounded interval [a,b], but I don't think it is on R. Could anyone please give me some hints how to prove it? 2.Fix 0 < η < 1. Suppose now...

An expression of the compactness theorem for sets of sentences is that: let T be a set of sentences in L. Then T has a model iff every finite subset of T has a model. Could anyone give me some hints how to prove this? The first direction is straightforward: every model of T is a model of...

Homework Statement Show that if h is continuous on [0, ∞) and uniformly continuous on [a, ∞), for some positive constant a, then h is uniformly continuous on [0, ∞). Homework Equations The Attempt at a Solution I'm thinking of using the epsilon-delta definition of continuity...

Thanks. What if I change Q into all non-positive rational numbers, then it has a maximum. Would that work? Also, that's only one l-structure. Could you give me some hints about the other two possible l-structure?

Thanks a lot for your help. I can only think of <Q, >>, which would make 1,2 true and 3 false. And I'm not sure that I've interpreted 3 correctly. Could you explain it a little bit more please?

1. Homework Statement Let L = {P } be a first-order language with a binary relation symbol P as only non-logical symbol. By exhibiting three suitable L-structures prove (informally) that no two of the following sentences logically implies the other (i) ∀x∀y∀z(P (x, y) → (P (y, z) → P (x...

Homework Statement 1.Find a function f : R → R which is discontinuous at the points of the set {1/n : n a positive integer} ∪ {0} but is continuous everywhere else. 2. Find a function g : R → R which is discontinuous at the points of the set {1/n : n a positive integer} but is continuous...

Homework Statement Find a sequence (an) of positive real numbers such the sum of an from 1 to infinity is convergent but the number of k such that a(k+1)>ak divided by n tends to 1 as n tends to infinity. Homework Equations The Attempt at a Solution I don't have a clue how to find...

thanks, I got it!

I can write it as the sum of (z^n)*(1+w^n+w^2n)/n!, n from 0 to infinity. But I'm still not sure how to simplify 1+w^n+w^2n from 1+w+w^2=0. Could you explain it in a bit more details? Thanks a lot!