Recent content by hth

Wait, let's go back to the second part here. a_n >= a_n+1 > 0, with an = 1/ns Shouldn't that be ∑|a_(n+1) - a_(n)| converges instead??

I left out some work, let me pick back up here: By the identity, 2 sin A sin B = cos (B-A) - cos (B + A), this becomes, 2sin(x/2)\sum from 1 to n of sin kx = (cos x/2 - cos 3x/2) + (cos 3x/2 + cos 5x/2) + ... + (cos (n-1/2)x - cos (n+1/2)x) = cos (x/2) - cos (n + 1/2)x.

Alright, here's my attempt at part 3. \sum sin(nx) = [(cos (x/2) - cos(n + 1/2)x) / (2sin(x/2))] for all x with sin(/2) =/= 0. We have, sin(x/2) \sum sin(nx) = sin(x/2) sinx + sin(x/2) sin 2x + ... + sin (x/2) sin(n) By the identity, 2 sin A sin B = cos (B-A) - cos (B + A), this...

What did I do wrong in steps 1 & 2?

Alright, so, So, i) Let (n+1)^(s) = n^s + sC1 x^(s-1)(1) + sC2 x^(s-2)(1)^2 + ... Now, assuming n ≥ 0 we get that n^s is a small part or 1st term of the right hand side of above expression is ≥ the left hand side. Note: The other part sC1x^(s-1)(1) + sC2 x^(s-2)(1)^2 + ... is positive...

Hi, thank you for replying. Here's a re-wording of the problem. Use the Dirichlet test to show that the infinite series from n=1 to infinity of sin(nx)/n^(s) is convergent for 0 less than s less than or equal to 1. Note: x is any real number.

Homework Statement Given the infinite series, \sum n=1 to \infty of sin(nx)/n^(s) is convergent for 0 < s <= 1. Homework Equations The Attempt at a Solution Let f_n(x) = sin(nx) and g_n(x) = 1/n^(s) i.) lim n-> \infty f_n = 0 I'm not sure how to show this formally. Specifically, for a...

Isn't that the solution to it if f_n(x) = x^(n)? How does that apply here?

Homework Statement Let fn(x) = 1/(nx+1) on (0,1) where x is a real number. Show this function does not converge uniformly. Homework Equations The Attempt at a Solution I know why it is not uniformly convergent. Even though fn(x) goes to zero monotonically on the interval (0,1), it's not...

Homework Statement Show that if \sumak converges, then \sum from k to ∞ of ak goes to zero as k goes to ∞. Homework Equations The Attempt at a Solution I'm not really sure how to go about this proof. But, this is my attempt, First I tried to show that \sumak is convergent. Let c be a...

Alright, I see it now. Ty. Also, I'm sorry if I wasn't supposed to hijack the OP's thread like this.

Yeah, it does. g'(u) =/= 0 for all u, so g isn't constant...

Here's my attempt. Assume g(ui) =/= g(uj). Let g'(x) exist on ui < x < uj. So, there exists some point, c, on ui < c < uj. Now, g(uj) - g(ui) = (uj - ui)g'(c) g'(c) = [g(uj) - g(ui)] / [(uj - ui)] Note that the LHS cannot equal zero because g(uj) =/= g(ui). So, g'(c)...

How would you apply MVT? I don't understand that approach.

Alright, here's my attempt. f(tx1, tx2, tx3,... txn)=t^s*f(x1, x2, x3,... xn) for all t Prove that the \sum from i=1 to n of xi * df/dxi (x1, x2, x3,... xn) = sf(x1, x2, x3,... xn). Proof. Let f = f (x1, x2, x3,...,xn). Then, by differentiating the function f(ty) = t^(s)f(y) by...