Coefficient of xn in e5x+exe2x is ……

e5x+exe2x=e3x+e-x =∑n=0∞ (3x)nn+∑n=0∞ (-x)nnxn coef=3nn+(-1)nn

Coefficient of xn in e5x+exe2x is ……

e5x+exe2x=e3x+e-x =∑n=0∞ (3x)nn+∑n=0∞ (-x)nnxn coef=3nn+(-1)nn

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Coefficient of xⁿ in $\frac{e^{5 x} + e^{x}}{e^{2 x}} is \dots \dots$

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$\frac{3^{n} + {(- 1)}^{n}}{n!}$

$\frac{1}{n!}$

$\frac{1}{n!} (3^{n} + 1)$

$\frac{3^{n} + 5^{n}}{n!}$

answer is C.

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Detailed Solution

$\begin{array}{rcl} \frac{e^{5 x} + e^{x}}{e^{2 x}} & = & e^{3 x} + e^{- x} \\ = & \sum_{n = 0}^{\infty} \frac{{(3 x)}^{n}}{n} + \sum_{n = 0}^{\infty} \frac{{(- x)}^{n}}{n} \\ x^{n} coef & = & \frac{3^{n}}{n} + \frac{{(- 1)}^{n}}{n} \end{array}$

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