MHB Does this series converge almost everywhere?

  • Thread starter Thread starter Jack3
  • Start date Start date
  • Tags Tags
    Set
Jack3
Messages
9
Reaction score
0
Show that ∑_(n=1)^∞ cos^n (2^n x) converges for a.e. x, but diverges on a dense set of x’s .

Show that [math]\sum_{n=1}^\infty \cos^n (2^n x)[/math] converges for a.e. x, but diverges on a dense set of x’s .
 
Last edited by a moderator:
Physics news on Phys.org
Re: Show that ∑_(n=1)^∞ cos^n (2^n x) converges for a.e. x, but diverges on a dense set of x’s .

Hi Jack,

Did you know that you can use Latex on MHB? The way you write is pretty close already to the correct Latex syntax so if you just learn a few common pieces of code you'll be able to use it immediately.

I rewrote the sum in your OP as:

\sum_{n=1}^\infty \cos^n (2^n x)

Jameson
 
Re: Show that ∑_(n=1)^∞ cos^n (2^n x) converges for a.e. x, but diverges on a dense set of x’s .

Jack said:
Show that [math]\sum_{n=1}^\infty \cos^n (2^n x)[/math] converges for a.e. x, but diverges on a dense set of x’s .
If $x$ is of the form $\dfrac{a\pi}{2^b}$ (where $a$ and $b$ are integers) then $\cos^n (2^n x)$ will take the value 1 infinitely often. That deals with showing that the series diverges on a dense set.

Convergence a.e. looks harder. I will pass on that for now.
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.

Similar threads

Replies
1
Views
1K
Replies
17
Views
1K
Replies
5
Views
2K
Replies
3
Views
2K
Replies
1
Views
1K
Replies
9
Views
2K
Replies
4
Views
2K
Back
Top