Stitz-Zeager_College_Algebra_e-book

# Using the properties of the factorial we have k 1 k

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Unformatted text preview: k−1 where k=0 k −1 k−1 ( 1) k − 1 ≥ 0. We get ck = 2(−−1)+1 = (−1) 1 , where now k ≥ 1. We leave to the reader to verify 2k − k that {ck }∞ generates the same list of numbers as does {bk }∞ , but the former satisﬁes Deﬁnition k=1 k=0 554 Sequences and the Binomial Theorem 9.1, while the latter does not. Like so many things in this text, we acknowledge that this point is pedantic and join the vast majority of authors who adopt a more relaxed view of Deﬁnition 9.1 to include any function which generates a list of numbers which can then be matched up with the natural numbers.2 Finally, we wish to note the sequences in parts 5 and 6 are examples of sequences described recursively. In each instance, an initial value of the sequence is given which is then followed by a recursion equation − a formula which enables us to use known terms of the sequence to determine other terms. The terms of the sequence in part 6 are given a special name: fn = n! is called n-factorial. Using the ‘!’ notation, we can desc...
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