*This preview shows
page 1. Sign up
to
view the full content.*

**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...

View
Full
Document