Way to transcendence Can both infinite, rational terms of series  and finite, irrational coefficients of polynomials  generate and eventually identify transcendental numbers? 
Identifying Trancendental Numbers ok the first link I would recommend is the work of the late Alan Baker: https://en.wikipedia.org/wiki/Baker%27s_theorem And a good paper by Michel Waldschmidt entitled "Transcendence of Periods: The State of the Art" that has a lot of good theorems for deciding whether a number is transcendental or not that follow from Baker's theorem of linear independence. This file size is too large for the limit on this forum, so if you cannot find a copy on the net, pm me and I will email you a copy. 
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Well sometimes the old ways are the best. But we can all rest easy now that a teenager with what appears to be a severe Adderall induced psychosis has shown us that simple elementary proof in a recent thread 
i^i is actually a real transcendental number. Here is how you can compute the value of i^i = 0.207879576... 1. Since e^(ix) = Cos x + i Sin x, then let x = Pi/2. 2. Then e^(iPi/2) = i = Cos Pi/2 + i Sin Pi/2; since Cos Pi/2 = Cos 90 deg. = 0. But Sin 90 = 1 and i Sin 90 deg. = (i)*(1) = i. 3. Therefore e^(iPi/2) = i. 4. Take the ith power of both sides, the right side being i^i and the left side = [e^(iPi/2)]^i = e^(Pi/2). 5. Therefore i^i = e^(Pi/2) = .207879576... From Cliff Pickover, The 15 Most Famous Transcendental Numbers  Cliff Pickover 
Baker's theorem deserves more contemplation. 
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I looked up Alan Baker, I'd never heard of him before. Won the Fields medal in 1970 for his work on transcendentals. I was fascinated to discover that it's unproven whether $\pi^e$ is transcendental. 
One great theorem that arises from Schwarz's Lemma in the work of Schneider on Abelian functions is this: Let a and b be rational numbers which are not integers and such that a + b is not an integer. Then B(a,b) is transcendental https://en.wikipedia.org/wiki/Beta_function 
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