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Question: In the expansion of \((e^{x} - 1)(e^{- x} + 1)\), the coefficient of \(x^{3}\) is....

In the expansion of (ex−1)(e−x+1)(e^{x} - 1)(e^{- x} + 1), the coefficient of x3x^{3} is.

A

0

B

1/3

C

2/3

D

1/6

Answer

1/3

Explanation

Solution

2(1x+13x3+15x5+...∞)2\left( \frac{1}{x} + \frac{1}{3x^{3}} + \frac{1}{5x^{5}} + ...\infty \right)

2(1x−13x3+15x5−...∞)2\left( \frac{1}{x} - \frac{1}{3x^{3}} + \frac{1}{5x^{5}} - ...\infty \right)

∴ The coefficient of log⁡e45+14−12(14)2+13(14)3+.....\log_{e}\frac{4}{5} + \frac{1}{4} - \frac{1}{2}\left( \frac{1}{4} \right)^{2} + \frac{1}{3}\left( \frac{1}{4} \right)^{3} + ......