In the expansion of (e^x - 1)(e^{-x} + 1),the | vedclass

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(B) Given expression: $(e^x - 1)(e^{-x} + 1)$$= (e^x - 1)\left(\frac{1 + e^x}{e^x}\right)$$= \frac{e^{2x} - 1}{e^x} = e^x - e^{-x}$Using the exponenti

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$1/3$

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(B) Given expression: $(e^x - 1)(e^{-x} + 1)$$= (e^x - 1)\left(\frac{1 + e^x}{e^x}\right)$$= \frac{e^{2x} - 1}{e^x} = e^x - e^{-x}$Using the exponential series expansion: $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$ and $e^{-x} = 1 - x + \frac{x^2}{2!} - \frac{x^3}{3!} + \dots$$e^x - e^{-x} = (1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots) - (1 - x + \frac{x^2}{2!} - \frac{x^3}{3!} + \dots)$$= 2x + 2\frac{x^3}{3!} + \dots$$= 2x + 2\frac{x^3}{6} + \dots = 2x + \frac{x^3}{3} + \dots$The coefficient of $x^3$ is $1/3$.

In the expansion of $(e^x - 1)(e^{-x} + 1)$,the coefficient of $x^3$ is

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