Discrete Mathematics

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Discrete Mathematics

WEN-CHING LIEN Department of Mathematics National Cheng Kung University

2008

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9.5: The summation operator

Definition

For f(x) =a₀+a₁x+a₂x²+a₃x³+ · · ·, the function f(x) 1−x has the following expansion:

f(x)

1−x = h

a₀+ +a₁x +a₂x²+ · · ·i h

1+x+x²+ · · ·i .

= a0+ (a0+a1)x+ (a0+a1+a2)x²+ · · · Here, 1

1−x is called the summation operator.

WEN-CHINGLIEN Discrete Mathematics

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Note:

1 1

1−x is the generating function for the sequence 1,1,1, ...;

hence, 1

(1−x)² is the generating function for the sequence 1,2,3, ...

2 x +x²is the generating function for the sequence 0,1,1,0,0,0, ...

3 x+x²

1−x is the generating function for the sequence 0,1,2,2,2, ...

4 x+x²

(1−x)² is the generating function for the sequence 0,1,3,5,7, ...

5 x+x²

(1−x)³ is the generating function for the sequence 0,1,4,9,16, ...

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Note:

1 1

hence, 1

3 x+x²

4 x+x²

5 x+x²

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Note:

1 1

hence, 1

3 x+x²

4 x+x²

5 x+x²

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Note:

1 1

hence, 1

3 x+x²

4 x+x²

5 x+x²

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Note:

1 1

hence, 1

3 x+x²

4 x+x²

5 x+x²

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Example

Find a formula to express 0²+1²+2²+ · · · +n²as a function of n.

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Discrete Mathematics

Discrete Mathematics

9.5: The summation operator

Thank you.