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The summation identities - acgusta2 - 10/25/2015
I was looking at the summation identity that is equal to e^x and from that I derived some other summation identities. [attachment=1212] 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. RE: The summation identities - tommy1729 - 10/25/2015
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 RE: The summation identities - acgusta2 - 10/26/2015
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 [attachment=1214] means. |