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The discussion centers on the definitions of hyperbolic cosine and its properties. The correct formula for cosh(2x) is cosh(2x) = (e^(2x) + e^(-2x)) / 2, not (e^(2x) - e^(-2x)) / 2, which actually corresponds to sinh(2x). Participants emphasize the importance of using parentheses for clarity in expressions involving exponents. There is also a mention of ensuring that reliable sources provide accurate mathematical information. Overall, the conversation clarifies the correct formulation of hyperbolic functions.
First, I tried to show that ##f_n## converges uniformly on ##[0,2\pi]##, which is true since ##f_n \rightarrow 0## for ##n \rightarrow \infty## and ##\sigma_n=\mathrm{sup}\left| \frac{\sin\left(\frac{n^2}{n+\frac 15}x\right)}{n^{x^2-3x+3}} \right| \leq \frac{1}{|n^{x^2-3x+3}|} \leq \frac{1}{n^{\frac 34}}\rightarrow 0##. I can't use neither Leibnitz's test nor Abel's test. For Dirichlet's test I would need to show, that ##\sin\left(\frac{n^2}{n+\frac 15}x \right)## has partialy bounded sums...
It's easy to see that any polynomial is function of that class. Also it seems that composition of exponent and polynomial is good as well. G_f(a, b) should be equal G_f(b, a) as the f(a+b) is symmetrical.
Does anyone have information about this or at least related to it?