First, apologies for the notation.. the computer I am currently using does not support the various buttons to create proper notation.(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

Given h(x)=|x| over [-1,1], extend the definition such that h(x+2)=h(x). That is, make h(x) be a periodic sawtooth function.

Now let h_{n}(x)=(1/2^{n})h(2^{n}x).

Then define g(x)=sigma[h_{n}(x)]=sigma[(1/2^{n})h(2^{n}x)] as n goes from 0 to infinity.

Define the sequence x_{m}=1/2^{m}, where m=0, 1, 2, ...

Prove that [g(x_{m})-g(0)]/[x_{m}-0]=m+1 and use this to prove that g'(0) does not exist.

2. Relevant equations

3. The attempt at a solution

I really have no idea. The best I have been able to do is to write the function as:

g(x)=sigma[(1/2^{n})|2^{n}x|] as n goes from 0 to infinity

and then substitute x_{m}for x to yield:

g(x)=sigma[(1/2^{n})|2^{n}*(1/2^{m})|] as n goes from 0 to infinity

and simplify it to:

g(x)=sigma[(1/2)^{n}|2^{n-m}|] as n goes from 0 to infinity. Otherwise, I have no idea.

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# Homework Help: Real Analysis Function defined as sum

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