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Summation by Sigma Method

Find the co-e...

Question

Find the co-efficient of $x^{7}$ in the series of $e^{2 x}$ .

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Solution

Expansion of $e^{x}$

$e^{x} = 1 + \frac{x}{1!} + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \frac{x^{4}}{4!} + . . .$

Expansion of $e^{2 x}$

$e^{2 x} = 1 + \frac{2 x}{1!} + \frac{4 x^{2}}{2!} + . . . . \frac{(2 x)^{7}}{7!}$

Co-efficient of $x^{7} = \frac{(2)^{7}}{7!}$

$= \frac{8}{315}$

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