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Let A(x) be the series A(x) lowing sequences in terms of compositions and sums involving the expression A(x) (rather than En>o (something). Here is an example: En>o anx". Express the generating series for each of the fol- 0, the nth term of the sequence is Sequence: — ао, а1, —а2, аз, —а4, а5, .... (–1)"+lan.) (Starting with n = Solution: The generating series for this sequence is given by: E(-1)"+'a anx" n>0 n20 = -En(-2)* n>0 = -A(-x) (а) 0, ао, 0, а1,0, а2, .... (Starting with n = 0, the nth term is 0 if n is even, and a: if n is odd.) (The 0th term is ao, and for all n > 1 the nth term is given (b) ао, ал + 2ао, а2 + 2ал, аз + 2а2, .... by an + 2an-1.)

Let A(x) be the series A(x) lowing sequences in terms of compositions and sums involving the expression A(x) (rather than En>o (something). Here is an example: En>o anx". Express the generating series for each of the fol- 0, the nth term of the sequence is Sequence: — ао, а1, —а2, аз, —а4, а5, .... (–1)"+lan.) (Starting with n = Solution: The generating series for this sequence is given by: E(-1)"+'a anx" n>0 n20 = -En(-2)* n>0 = -A(-x) (а) 0, ао, 0, а1,0, а2, .... (Starting with n = 0, the nth term is 0 if n is even, and a: if n is odd.) (The 0th term is ao, and for all n > 1 the nth term is given (b) ао, ал + 2ао, а2 + 2ал, аз + 2а2, .... by an + 2an-1.)

Advanced Engineering Mathematics
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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1. Let A(x) be the series A(x) lowing sequences in terms of compositions and sums involving the expression A(x) (rather than En>o (something). Here is an example: En>o anx". Express the generating series for each of the fol- (Starting with n = 0, the nth term of the sequence is Sequence: -ao, a1, –a2, a3, –a4, a5,.…... (-1)n+lan.) Solution: The generating series for this sequence is given by: E(-1)"+la, *anx" n>0 n>0 = -E an(-x)" n20 = -A(-x) (а) 0, ао, 0, а1, 0, а2, .... (Starting with n = 0, the nth term is 0 if n is even, and a n-1 if n is odd.) (The 0th term is ao, and for all n > 1 the nth term is given (b) ао, а1 + 2ао, а2 + 2а1, аз + 2а2, .... by an + 2an-1.)
Transcribed Image Text:1. Let A(x) be the series A(x) lowing sequences in terms of compositions and sums involving the expression A(x) (rather than En>o (something). Here is an example: En>o anx". Express the generating series for each of the fol- (Starting with n = 0, the nth term of the sequence is Sequence: -ao, a1, –a2, a3, –a4, a5,.…... (-1)n+lan.) Solution: The generating series for this sequence is given by: E(-1)"+la, *anx" n>0 n>0 = -E an(-x)" n20 = -A(-x) (а) 0, ао, 0, а1, 0, а2, .... (Starting with n = 0, the nth term is 0 if n is even, and a n-1 if n is odd.) (The 0th term is ao, and for all n > 1 the nth term is given (b) ао, а1 + 2ао, а2 + 2а1, аз + 2а2, .... by an + 2an-1.)
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