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**Unformatted text preview: **= n−1 2n−1
2n−1 , , n≥1 n≥1 (g) geometric, r = 10
(h) neither
n
, n≥1
3n−1
1
an = n2 , n ≥ 1
(−1)n−1 x2n−1
, n≥
2n−1 10n −1
10n , (d) an = (g) an = (e) (h) an = (n + 2)3 , n ≥ 1 (f) 1 (i) an = n≥1 1+(−1)n−1
,
2 n≥1 562 9.2 Sequences and the Binomial Theorem Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems
concerning the sum of terms of a sequence. We begin with a deﬁnition, which, while intimidating,
is meant to make our lives easier.
Definition 9.3. Summation Notation: Given a sequence {an }∞ k and numbers m and p
n=
satisfying k ≤ m ≤ p, the summation from m to p of the sequence {an } is written
p an = am + am+1 + . . . + ap
n=m The variable n is called the index of summation. The number m is called the lower limit of
summation while the number p is called the upper limit of summation.
In English, Deﬁnition 9.3 is simply deﬁning a short-hand notation for adding up the terms of the
sequence {an }∞ k from am through ap . The symbol Σ is the capi...

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