In the expansion of frac{e^{4x} - 1}{e^{2x}}, | vedclass

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(C) We are given the expression $\frac{e^{4x} - 1}{e^{2x}}$.Dividing the terms,we get $\frac{e^{4x}}{e^{2x}} - \frac{1}{e^{2x}} = e^{2x} - e^{-2x}$.Us

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(C) We are given the expression $\frac{e^{4x} - 1}{e^{2x}}$.Dividing the terms,we get $\frac{e^{4x}}{e^{2x}} - \frac{1}{e^{2x}} = e^{2x} - e^{-2x}$.Using the exponential series expansion $e^t = 1 + t + \frac{t^2}{2!} + \frac{t^3}{3!} + \dots$,we have:$e^{2x} = 1 + (2x) + \frac{(2x)^2}{2!} + \frac{(2x)^3}{3!} + \dots = 1 + 2x + 2x^2 + \frac{4x^3}{3} + \dots$$e^{-2x} = 1 + (-2x) + \frac{(-2x)^2}{2!} + \frac{(-2x)^3}{3!} + \dots = 1 - 2x + 2x^2 - \frac{4x^3}{3} + \dots$Subtracting these series: $(e^{2x} - e^{-2x}) = (1 - 1) + (2x - (-2x)) + (2x^2 - 2x^2) + \dots = 4x + 0x^2 + \frac{8x^3}{3} + \dots$The coefficient of $x^2$ is $0$.

In the expansion of $\frac{e^{4x} - 1}{e^{2x}}$,the coefficient of $x^2$ is

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