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Lecture 6
Sequences
Math 13200
Definition
Recall that N is the set of “natural numbers,” i.e. the positive integers, while R is the set of real
numbers (i.e., the set of numbers that we have been dealing with all year - it includes both rational
numbers and irrational numbers).
We’ve been studying functions f from R to R all year. I.e., a rule which assigns for each real number
x a real number f (x). Modern mathematicians write this as:
f: R→R
x 7→ f (x)
The name of the function is f , the domain is R, the target is R, and f takes x to f (x).
A sequence is a function with domain N and target R. That is, it is a rule which associates with
each positive integer n a real number a(n), sometimes written an .
a:N→R
n 7→ an
(Interpretation of the above line: the function name is a, the source is N, the target is R, and a
takes n to an .)
There are several ways to describe a sequence. One way is to give enough terms to establish an
imlied pattern (this is not very rigorous, but used often):
1, 2, 3, 4, 5, . . .
1, 4, 7, 10, 13, . . .
2, 5, 10, 17, 26, . . .
1, 1, 2, 3, 5, 8, 13, 21 . . .
Alternatively, one can give a recursion formula which explains how to derive an from the preceding
numbers in the sequence
a1 = 1
an = an−1 + 1
a1 = 1
an = an−1 + 3
a1 = 2
a1 = a2 = 1
an = an−1 + 2n − 1
an = an−1 + an−2
1
Finally, one can give an explicit formula for the nth term of the sequence (similar to how we
often give explicit formulas for functions of real numbers):
an = n
an = 3n − 2
an = n2 + 1
√
√
(1 + 5)n − (1 − 5)n
√
an =
2n 5
(You can check that these collections of sequences are all the same!)
Geometric Picture
There are two fruitful ways to view sequences geometrically:
• By plotting them directly on the number line
• By considering their graphs
Let’s draw these two pictures for several sequences:
1. an = (−1)n +
2. an = 1 −
1
n
1
n
3. an = .999
4. an = 1 + (0.5) + (0.5)2 + · · · + (0.5)n
Convergence
Consider the first 3 sequence from the previous section. They all have values that “bunch up” near
1. But which one(s) converge to 1? Just the second one! Does the 4th sequence converge? Yes to 1/(1 − 0.5) = 2.
The value that a sequence converges to is called the limit of the sequence, and is denoted
lim an
n→∞
(Read last paragraph of page 449 of the book.)
There exists a rigorous − L definition of the limit of a sequence, i.e. the value that the sequence
converges to, in the book. It is in fact very similar to that for a function. Indeed, your intuition for
limits of sequences should be the same as for infinite limits (i.e. limx→∞ ) for ordinary functions.
See the below picture (from the book):
2
Limit Properties
Suppose an and bn are convergent sequences, and k is a constant. Then:
(i) lim k = k
n→∞
(ii) lim kan = k · lim an
n→∞
n→∞
(iii) lim (an ± bn ) = lim an ± lim bn
n→∞
n→∞
n→∞
(iv) lim (an · bn ) = lim an · lim bn
n→∞
n→∞
n→∞
lim an
an
= n→∞ , provided that the limit of bn is nonzero.
n→∞ bn
lim bn
(v) lim
n→∞
(vi) If f (x) is a function such that an = f (n), and lim f (x) exists, then
x→∞
lim an = lim f (x)
n→∞
x→∞
Examples
1/np , where p is a positive integer.
3n2 /(7n2 + 1)
an converges when |a| < 1.
(−1)n
n2
2n2 + 3
3