MHB Does this series converge almost everywhere?

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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 .
 
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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.
 
I posted this question on math-stackexchange but apparently I asked something stupid and I was downvoted. I still don't have an answer to my question so I hope someone in here can help me or at least explain me why I am asking something stupid. I started studying Complex Analysis and came upon the following theorem which is a direct consequence of the Cauchy-Goursat theorem: Let ##f:D\to\mathbb{C}## be an anlytic function over a simply connected region ##D##. If ##a## and ##z## are part of...

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