MHB Kummer's Theorem: An Elementary Proof?

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Kummer's Theorem is the focus of the discussion, with participants seeking an elementary proof. A recommended resource for understanding hypergeometric series is the book "A = B" by Marko Petkovsek, Herbert S. Wilf, and Doron Zeilberger, which is available for free download. The conversation highlights the interest in finding simpler proofs for complex mathematical theorems. Participants express enthusiasm about exploring the suggested material. The discussion emphasizes the quest for accessible mathematical proofs.
alyafey22
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I came across the interesting Kummer's Theorem .Does anybody know an elementary proof ?
 
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Thanks , I'll take a look at it :D
 

Thread 'Proving that convexity implies second order derivative being positive'

There are probably loads of proofs of this online, but I do not want to cheat. Here is my attempt: Convexity says that $$f(\lambda a + (1-\lambda)b) \leq \lambda f(a) + (1-\lambda) f(b)$$ $$f(b + \lambda(a-b)) \leq f(b) + \lambda (f(a) - f(b))$$ We know from the intermediate value theorem that there exists a ##c \in (b,a)## such that $$\frac{f(a) - f(b)}{a-b} = f'(c).$$ Hence $$f(b + \lambda(a-b)) \leq f(b) + \lambda (a - b) f'(c))$$ $$\frac{f(b + \lambda(a-b)) - f(b)}{\lambda(a-b)}...
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