- Thread starter
- #1

- Thread starter Jack
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- Thread starter
- #1

- Admin
- #2

- Jan 26, 2012

- 4,043

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

- Moderator
- #3

- Feb 7, 2012

- 2,706

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.Show that \(\displaystyle \sum_{n=1}^\infty \cos^n (2^n x)\) converges for a.e. x, but diverges on a dense set of x’s .

Convergence a.e. looks harder. I will pass on that for now.