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 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...
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...
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!
Homework Statement
Let ω be the complex number e^(2πi/3)， Find the power series for e^z + e^(ωz) + e^((ω^2) z).
Homework Equations
The Attempt at a Solution
I can show that 1+w+w^2=0, don't know if it would help. Could anyone please give me some hints? Any input is appreciated!
There are five statements:
(a) If n^2 an → 0 as n → ∞ then ∑ an converges.
(b) If n an→ 0 as n → ∞ then ∑an converges.
(c) If ∑an converges, then ∑((an )^2)converges.
(d) If ∑ an converges absolutely, then ∑((an )^2) converges.
(e) If ∑an converges absolutely, then |an | < 1/n for all...
1. (1) (a) Let X be a topological space. Prove that the set Homeo(X) of home-
omorphisms f : X → X becomes a group when endowed with the binary operation f ◦ g.
(b) Let G be a subgroup of Homeo(X). Prove that the relation ‘xRG y ⇔ ∃g ∈ G such
that g(x) = y ’ is an equivalence relation.
(c)...
thanks for the reply, but I'm still not sure about which non-convergent sequence to choose, would something like 1/n work? I just don't know how to use the complete norm here.
Homework Statement
Let l∞ be the space of bounded sequences of real numbers, endowed with the norm
∥x∥∞ = supn∈N |xn | , where x = (xn )n∈N .
Prove that the closed unit ball of l∞ , B(0, 1) = {x ∈ l∞ ; ∥x∥∞ ≤ 1} , is not compact.
Homework Equations
The Attempt at a Solution
I'm...
1.Let R be a ring such that Z ⊂ R ⊂ Q. Show that R is a principal ideal domain.
We show that Z is a principal ideal domain, so every ideal in Z which is also in R is principal. But I'm not sure how to use that R is contained in Q.
2. Proof that X^4+1 is reducible in Z/pZ [X] for every...
(a) Let R and S be rings with groups of units R∗ and S ∗ respectively. Prove that
(R × S)∗ = R∗ × S ∗ .
(b) Prove that the group of units of Zn consists of all cosets of k with k coprime to n.
Denote the order of (Zn )∗ by φ(n); this is Euler’s φ-function.
(c) Now suppose that m and n are...
Let I be an open interval in R and let f : I → R be a diﬀerentiable function.
Let g : T → R be the function deﬁned by g(x, y) =(f (x)−f (y))/(x-y)
1.Prove that g(T ) ⊂ f (I) ⊂ g(T ) (The last one should be the closure of g(T), but I can't type it here)
2. Show that f ′ (I) is an interval...
Let A, B be two connected subsets of a topological space X such that A intersects the closure of B .
Prove that A ∪ B is connected.
I can prove that the union of A and the closure of B is connected, but I don't know what to do next. Could anyone give me some hints or is there another way to...
Homework Statement
We have x^2+y^2+z^2=1, and we want to find the maximum and minimum value of xz+xy-yz.
Homework Equations
The Attempt at a Solution
I've simplified the original problem to this point, but I'm not sure what to do next. Could anyone give me some hints? Any help is...
Homework Statement
1.Prove that if A is a real matrix then At A is diagonalisable.
2. Given a known 3*3 matrix A, Calculate the maximum and minimum values of ||Ax|| on the sphere ||x|| = 1.
Homework Equations
The Attempt at a Solution
For the first problem, I'm thinking of...