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Show that for small positive $x$, $$\left( \sin x \right)^{\cos x} = x -\left( 3 \log x + 1\right) \frac{x^{3}}{3!} + \Big( 15 \log^{2} x + 15 \log x + 11 \Big) \frac{x^{5}}{5!} + \mathcal{O}(x^{7})$$
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A series expansion is a mathematical tool used to approximate a function using a finite number of terms. It is particularly useful for functions that cannot be easily integrated or differentiated.
The purpose of expanding Sin^Cos x is to obtain a more simplified and manageable expression for the function. This can help with calculations and analysis of the function.
The expansion of Sin^Cos x is useful because it allows us to approximate the function with a polynomial, which is a much simpler form to work with. It also helps us to understand the behavior of the function and make predictions about its values.
The notation "O(x^3)" represents the order of the terms in the expansion. It indicates that the terms in the expansion become increasingly smaller as x approaches 0, and the terms beyond the third term can be neglected without significantly affecting the overall accuracy of the approximation.
No, the series expansion is only an approximation of Sin^Cos x. As more terms are added to the expansion, the approximation becomes more accurate, but it will never be an exact representation of the function.