The coefficient of x^n in the expansion of fr | vedclass

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(D) Given the expression $\frac{e^{7x} + e^x}{e^{3x}}$.Dividing the terms,we get $\frac{e^{7x}}{e^{3x}} + \frac{e^x}{e^{3x}} = e^{4x} + e^{-2x}$.Using

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$\frac{4^n + (-2)^n}{n!}$

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(D) Given the expression $\frac{e^{7x} + e^x}{e^{3x}}$.Dividing the terms,we get $\frac{e^{7x}}{e^{3x}} + \frac{e^x}{e^{3x}} = e^{4x} + e^{-2x}$.Using the expansion $e^y = \sum_{n=0}^{\infty} \frac{y^n}{n!}$,we have:$e^{4x} = \sum_{n=0}^{\infty} \frac{(4x)^n}{n!} = \sum_{n=0}^{\infty} \frac{4^n x^n}{n!}$$e^{-2x} = \sum_{n=0}^{\infty} \frac{(-2x)^n}{n!} = \sum_{n=0}^{\infty} \frac{(-2)^n x^n}{n!}$Adding these,the coefficient of $x^n$ is $\frac{4^n + (-2)^n}{n!}$.

The coefficient of $x^n$ in the expansion of $\frac{e^{7x} + e^x}{e^{3x}}$ is

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