Sequences

I.  Definition of a Sequence

A sequence is a list of numbers, or more formally, a function f(n) from the natural numbers to the real numbers.

We write

a_n to mean the n^th term of the sequence.

Example:

If a_n = 1/(n + 2) then we have

a₁ = 1/3, a₂ = 1/4, etc.

Exercise:  

Write the general term a_n for the following sequences:

A)  -1,1,-1,1,-1,1,...

B)  1,4,9,16,...

C)  1/2, -1/6, 1/24,-1/120,...

D)  1, 1/2, -1/4, -1/8, 1/16, 1/32, -1/64, -1/128, ...

II.  The Limit of a Sequence

Consider the sequence

1/2, 2/3, 3/4, 4/5, ...

We see that as n becomes large the numbers approach 1.  In particular if you give me any small error number e, I can find an N such that all a_n for n > N, |a_n -1| < e.  We say that the limit of the sequence approaches 1.  

In general, if a_n is a sequence that converges to a limit L then for any e > 0, we can find an N such that for all n > N

|a_n - L| < e  

If there is no such L then we say that the sequence diverges.

Theorem:  Let f(n) = a_n  be a sequence, then a_n -> L if and only if lim f(n) as n -> infinity = L.

Example:  We find the limit of the sequence a_n = (2n + 1)/(n-3) by considering the function

f(n) = (2n + 1)/(n - 3)  We not that as n -> infinity, we get infinity/infinity hence we can use L'Hopital's Rule:  Taking derivatives of the top and bottom, we have

2/1 hence the limit is 2.

III.  The Squeeze Theorem
Suppose that lim a_n = lim b_n = L

and that there is an N such that for any n > N,

a_n  <  c_n  <  b_n

then lim c_n = L

Example

We will show that the lim of sin(n)/n! converges to 0.

Note that

-1/n < sin(n)/n! < 1/n  

both the left hand and right hand sides converge to 0 hence sin(n)/n! converges to 0 also.

IV.  Monotonic and Bounded Sequences

A sequence is monotonically increasing (decreasing) if

a_n > a_n+1 (a_n < a_n+1 ) for all n.

A sequence is bounded from above (below) if there is a number M such that

a_n < M ( a_n > M) for all n.

Exercises:  

Classify the monotonicity and boundedness of the following sequences:

A)  a_n = sin(n)

B)  a_n = 1/n

C)  a_n  = (n + 1)/(n + 2)

D)  a_n = n² + 1 

Theorem:  A bounded monotonic sequence converges