1. Sequences (Sec.11.2) A sequence is an infinite list of numbers
a1, a2, a3, a4, …, ordered by the natural numbers (n) 1, 2, 3,…
Examples:
If {an} is a sequence, call an the nth term. A sequence is also written
For above examples

2. Limits
Let be a sequence. Then
means
an approaches as n becomes larger and larger.
Convention: If sequence has limit, we say that an converges to and that the sequence is convergent – otherwise the sequence is divergent.

3. Examples

4. Theorem: If lim x f(x) = L and an = f(n), n
then lim n an = L.
Example: Evaluate
Sol:

5. Also the squeeze law applies:
Limit Laws
Same limit theorems hold as for functions. Thus
if an and bn are convergent sequences and
lim n an = L , lim n bn = M
then
lim n (an + bn ) = L + M,
lim n an bn = LM,
lim n an / bn = L/ M provided M 0.
Also the squeeze law applies:
If an, bn  cn for all n and
lim n an = L = lim n cn then lim n bn = L

6. Example
Find
USEFUL SEQUENCES TO REMEMBER

7. By squeeze law  lim n (2n + 1)1/n = 2
Example
Evaluate
2n  2n + 1  2n + 2  2. 2n, for all n 
 2  (2n + 1)1/n  21/n.2
By (2) above
lim n 21/n.2 = 2 lim n 21/n = 2
By squeeze law  lim n (2n + 1)1/n = 2

8. Series The sum of sequence is called a series, written
Call an the nth term. Add the first n terms:
Call sn the nth partial sum.

9. Set
We say the series converges to sum S if sn converges to a real number S.
Otherwise we say the series diverges.

10. So to find
start adding the terms one at a time, starting with the first term a1.
if sum approach some number, S, then S is the infinite sum.
Note:

11. Examples

14. The n th term test (Useful to show a series diverges)

15. Example

16. Geometric Series: Consider series
Sequence of partial sums is given by

19. Example

20. Comparison test
If 0  an  bn and converges then converges.
Examples