Undergrad Taylor Expansion Question about this Series

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The series presented is a Taylor expansion of the function f around the point x. It is expressed as f(x + α) and is a power series in α, indicating that the expansion is centered at x. Clarification is needed regarding the evaluation of derivatives, which should be explicitly calculated at the point x for better understanding. This will help in comprehending how the series approximates the function f near that point. The discussion emphasizes the importance of recognizing the center of the expansion in Taylor series.
LagrangeEuler
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Can you please explain this series
f(x+\alpha)=\sum^{\infty}_{n=0}\frac{\alpha^n}{n!}\frac{d^nf}{dx^n}
I am confused. Around which point is this Taylor series?
 
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LagrangeEuler said:
Can you please explain this series
f(x+\alpha)=\sum^{\infty}_{n=0}\frac{\alpha^n}{n!}\frac{d^nf}{dx^n}
I am confused. Around which point is this Taylor series?

THis is an expansion about x. You can tell that because the series is a power series in \alpha.
 
It would help if the derivatives were explicitly evaluated at ##x##. Then it would be clearer.
 

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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