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Sequences Section 8. A sequence is a function whose domain is the set of positive integers. We use the following notation and terminology.
1, 2, 3, 4, …, n, …, ↓ ↓ ↓ ↓ ↓ a 1 a 2 a 3 a 4 …, a (^) n …,
The numbers a1,a 2 , a 3 , … are called the terms of the sequence and a (^) n is called the nth term. We denote the sequence a1,a 2 , a 3 , … by a (^) n or a (^) n n^ ^1. List the terms of the sequence: 1. a (^) n 3 − 1 n^ 2. b (^) n n
2 2 n^ − 1 3. Recursively defined sequences: a. a (^) n 1 3 a (^) n, a 1 2 b. c (^) n 2 c (^) n 1 − c (^) n, c 1 1, c 2 3 Some special sequences: The arithmetic sequence: Successive terms found by adding a constant d: a (^) n a 1 n − 1 d.
The geometric sequence: Successive terms found by multiplying a constant r: g (^) n g 1 r n−^1 .
Find the nth term of the sequence: 1. −4, 5, 14, 23, 32, … 2. 1, −1, 1, −1, 1, … 3. 1, 12 , 14 , 18 , 161 , …
Find a recursive definition for the sequence: (^) 1, 1, 2, 3, 5, 8, 13, 21, 34, … (^)
Limit of a sequence : The limit of a sequence (^) a (^) n is a real number L denoted limn→ a (^) n L
If we can take the terms a (^) n as close to L as we desire by making n sufficiently large enough. If this limit exists, we say the sequence converges. If the limit does not exist, we say the sequences diverges.
A very useful Theorem : If f is a function such that limx→ fx L and fn a (^) n for every
positive integer n, then
limn→ a (^) n L
In other words, we do not have to reinvent the wheel trying to find limits of sequences, we can use all the tools we learned in finding limits of functions at infinity (Section 2.5 in text & L’Hôpital’s Rule). All the following properties of limits of sequences come directly from this last theorem. Let limn→ a (^) n L, limn→ b (^) n M, and c be a constant, then
- limn→ca (^) n cL 2. limn→a (^) n b (^) n L M
- limn→a (^) n b (^) n LM 4. limn→^ a^ n b (^) n
L
M
, b (^) n ≠ 0 & M ≠ 0
Determine whether or not the following sequences converge or diverge. If they converge, find their limit.
1. a (^) n (^) n^1 r , r 0 2. a (^) n 2 3. a (^) n 3 − 1 n^ 4. The sequence with nth term p (^) n 3 n
(^3) − 4 n (^2) 3 1 − 2 n − 5 n^3 5. The sequence with nth term a (^) n 3 n^ −^2 n^2 4 6. The sequence with nth term b (^) n 1 (^1) n n
7. The sequence with nth term c (^) n n
2 2 n^ − 1 8. The sequence with nth term d (^) n 2 n 1 − 2 n − 2
Yet another useful Theorem : The Squeeze Theorem for Sequences. If there is an integer N such that a (^) n ≤ c (^) n ≤ b (^) n for n N and limn→ a (^) n limn→ b (^) n L, then
limn→ c (^) n L Note: If limn→|a (^) n | 0, then since −|a (^) n | ≤ a (^) n ≤ |a (^) n | this theorem says limn→ a (^) n 0
Apply this to the following: Evaluate limn→
− 1 n n if it exists.
A sequence is monotonic if its terms are nondecreasing or nonincreasing.
Determine which sequences are monotonic. 1. a (^) n 3 − 1 n 2. b (^) n 2 n 1 n
A sequence a (^) n is bounded above if there is a number M such that a (^) n ≤ M for all n. A sequence a (^) n is bounded below if there is a number m such that a (^) n ≥ m for all n. A sequence a (^) n is bounded if it is bounded above and bounded below.
One Final Theorem : Every bounded, monotonic sequence is convergent.
Can you think of a sequence that is monotonic but not convergent? How about one that is bounded but not convergent?
