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The summation identities
#1
I was looking at the summation identity that is equal to e^x and from that I derived some other summation identities.

   

I used the summation identities that is equal to a^x to derive some summation identities that are equal to different size power towers. I found that the summation identities for power towers or tetration involve summations within summations and sometimes summations within summations within summations. I'm wondering if there would be a simpler way to express the summations within summations as if there is then I'm thinking it could help with tetration involving numbers other than the positive integers.
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#2
Question 
Welcome to the forum acgusta2.

Thank you for participating and using Tex.

However , I think you made a typo , I think after closing Every Bracket there should be a " ^k " , till the power k , right ?

Second , have you seen this :

http://mathworld.wolfram.com/PowerTower.html

It probably contains the formula you Search for ?

Its a Nice idea you have , but assuming the Above ; already investigated and nothing new.

Correct if im wrong.

Also , it is standard practice in math too use as many variables as possible IF it clarifies.
Ok that sounds cryptic , what I mean in this situation is :

You should Sum over k_1,k_2,... Otherwise the nested sums are confusing.
( Sum when over what , in what order etc )


It might seem irrelevant advice , but really it is crucial for the clarity of communication , and keep in Mind being clear reduces the probability of confusing yourself , in particular when you look at it again MUCH LATER.

Regards

Tommy1729
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#3
I edited my original post and changed some of the ks to k_1, k_2, and k_3 and also raised the nested sums to powers of k and found that if I added one more layer of nested sums I could produce self similarity for all layers.

I also saw the formula in the link from Wolfram, and I'm confused as to what

   

means.
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