11. Explain how we know that (1 – x)? is the generating function for 1, 2, 3, 4, ....
11. Explain how we know that (1 – x)? is the generating function for 1, 2, 3, 4, ....
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Transcribed Image Text:11. Explain how we know that
(1 – x)?
is the generating function for
1, 2, 3, 4, ....
Expert Solution
Step 1
The general expansion of the binomial (1-x)-1 will be,
(1-x)-1=1+x+x2+x3+….
Step 2
Assume the generating function of the given series be A.
A= 1+2x+3x2+4x3+….
Multiply by x.
Ax=x+2x2+3x3+4x4+…
Subtract the two equations.
A-Ax=1+x+x2+x3+…
From general expansion of binomial.
A(1-x)=(1-x)-1
A=(1-x)-2
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