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The Gelfond-Schneider Constant 2

^{21/2}is transcendental. Of my understanding, transcendental numbers are a special case of irrational numbers in the sense of how they may or may not be derrived. This being the case, how is it that many infinite series are stated with such confidence to be transcendental series. For example Sin(x) or Cos(x).

Furthermore, I don't require a lecture on it, but some material for reasearch would be appreciated - how does one prove what an infinite series converges to. I understand this is dependent on the series itself but I'm interested in proving more than the fact that the series converges or diverges.

Lastly, 2

^{21/2}being transcendental - are there any such symbolic representations of Pi or e

^{x}?

Danke, Kueffner