The coefficient of x^r in the expansion of 1 | vedclass

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Question

(C) The given series is the expansion of the exponential function $e^{a + bx}$.$S = e^{a + bx} = e^a \cdot e^{bx}$.Using the expansion $e^{bx} = 1 + \

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$\frac{e^a b^r}{r!}$

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(C) The given series is the expansion of the exponential function $e^{a + bx}$.$S = e^{a + bx} = e^a \cdot e^{bx}$.Using the expansion $e^{bx} = 1 + \frac{bx}{1!} + \frac{(bx)^2}{2!} + \dots + \frac{(bx)^r}{r!} + \dots$,we have:$S = e^a \left( 1 + \frac{bx}{1!} + \frac{b^2 x^2}{2!} + \dots + \frac{b^r x^r}{r!} + \dots \right)$.The coefficient of $x^r$ is obtained by multiplying $e^a$ with the term $\frac{b^r x^r}{r!}$.Thus,the coefficient of $x^r$ is $\frac{e^a b^r}{r!}$.

The coefficient of $x^r$ in the expansion of $1 + \frac{a + bx}{1!} + \frac{(a + bx)^2}{2!} + \dots + \frac{(a + bx)^n}{n!} + \dots$ is

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