MHB Need help on Fourier Series (badly)

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The discussion centers around a request for assistance with Fourier series, highlighting the user's struggle with specific questions. The user expresses frustration and urgency in seeking help, indicating that they find the topic particularly challenging. Participants are encouraged to share what progress has been made to facilitate more targeted assistance. The conversation emphasizes the importance of collaboration in tackling difficult mathematical concepts. Overall, the thread underscores the need for support in understanding Fourier series.
jimmy93
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Need help on Fourier series! Been stuck on this questions, it is too tough for me!

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jimmy93 said:
Need help on Fourier series! Been stuck on this questions, it is too tough for me!
What have you been able to do so far?

-Dan
 

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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