Book edition 7th

Author(s) Kenneth H. Rosen

Pages 808 pages

ISBN 9780073383095

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Short Answer

Find the sequence with each of these functions as its exponential generating function $f (x) = e^{3 x} - 3 e^{2 x}$ .

$a_{k} = 3^{k} - 3 \cdot 2^{k}$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given data

The given function is $f (x) = e^{3 x} - 3 e^{2 x}$

Step 2: Concept used of generating function

The ordinary generating function of a sequence $a_{n}$ is

$G (a_{n}; x) = \sum_{n = 0}^{\infty} a_{n} x^{n} .$

Step 3: Solve the function

Using $\sum_{k = 0}^{+ \infty} \frac{a_{k}}{k!} x^{k} = e^{x}$ , we have

$e^{3 x} - 3 e^{2 x}$ = $\sum_{k = 0}^{+ \infty} \frac{{(3 x)}^{k}}{k!} - 3 \sum_{k = 0}^{+ \infty} \frac{{(2 x)}^{k}}{k!}$

$= \sum_{k = 0}^{+ \infty} 3^{k} \frac{x^{k}}{k!} - 3 \sum_{k = 0}^{+ \infty} 2^{k} \frac{x^{k}}{k!}$

$= \sum_{k = 0}^{+ \infty} (3^{k} - 3 \cdot 2^{k}) \frac{x^{k}}{k!}$

The sequence $\{a_{k}\}$ are then the coefficients of $\frac{x^{k}}{k!}$ in the above sum: $a_{k} = 3^{k} - 3 \cdot 2^{k}$ .

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Discrete Mathematics and its Applications

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Short Answer

Find the sequence with each of these functions as its exponential generating function $f (x) = e^{3 x} - 3 e^{2 x}$ .

Step by Step Solution

Step 1: Given data

Step 2: Concept used of generating function

Step 3: Solve the function

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Discrete Mathematics and its Applications

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Find the sequence with each of these functions as its exponential generating function $f (x) = e^{3 x} - 3 e^{2 x}$ .