Word Count : 265

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To expand \( e^{2x} \) in powers of \( (x-1) \), we can use a Taylor series expansion. The general form of the Taylor series for a function \( f(x) \) around a point \( a \) is: \[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f^{(3)}(a)}{3!}(x-a)^3 + \cdots \] Here, we need to expand \( e^{2x} \) around \( x = 1 \). Let’s ____ __________ __________ ___ ___ ________ ______ _____ ______.
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