Chapter 10.
Infinite sequences and series
Section 10.1
Sequences
A sequence is a list of numbers written in a definite order: a
1
, a
2
, ..., a n
, ...
Definition A sequence { a n
} has the limit L and we write lim n →∞ a n
= L or a n
→ L as n → ∞ if we can make the terms a n as close to L as we like by taking n sufficiently large.
If lim n →∞ a n exists, we say the sequence sequence diverges or is divergent converges or is convergent . Otherwise, we say the
Limit Laws If { a n
} and { b n
} are convergent sequences and c is a constant, then
1. lim n →∞
[ a n
+ b n
] = lim n →∞ a n
+ lim n →∞ b n
2. lim n →∞
[ a n
− b n
] = lim n →∞ a n
− lim n →∞ b n
3. lim x →∞ ca n
= c lim n →∞ a n
4. lim n →∞
[ a n b n
] = lim n →∞ a n
· lim n →∞ b n
5. lim n →∞ a n b n
= lim n →∞ a n lim n →∞ b n if lim n →∞ b n
= 0
6. lim n →∞ c = c
The Squeeze Theorem If a n
≤ b n
≤ c n for n ≥ n
0 and lim n →∞ a n
= lim n →∞ c n
= L , then lim n →∞ b n
= L .
Theorem If lim n →∞
| a n
| = 0, then lim n →∞ a n
= 0.
Example 1.
Find the limit n 2
(a) lim n →∞
( − 1) n
1 + n 3
1

(b) lim n →∞ cos
2 n
2 n
π n
(c) lim n →∞ 3 n
Definition decreasing if a
A sequence n
> a n +1
{ a n
} is called increasing if a n
< a n +1 for all n ≥ 1. It is called for all n ≥ 1. A sequence is monotonic if it is either increasing or decreasing.
Example 2.
Determine whether the sequence is increasing, decreasing, or not monotonic.
1
(a) a n
=
3 n + 5
2

(b) a n
= 3 +
( − 1) n n
(c) a n
= n − 2 n + 2
Definition A sequence { a n
} is bounded above if there is a number M such that a n
≤ M for all n ≥ 1
It is bounded below if there is a number m such that a n
≥ m for all n ≥ 1
If it is bounded above and below, then { a n
} is a bounded sequence
Monotonic Sequence Theorem Every bounded, monotonic sequence is convergent.
Example 3.
Show that the sequence defined by
1 a
1
= 2 a n +1
=
3 − a n
3

is decreasing and bounded. Find the limit of { a n
} .
4