some basics
A sequence {an }+1
= a0 , a1 , a2 , . . . is = converges to L if
0
lim an = L
n!+1
More formally, a sequence {an }+1
converges to L if, = given an
0
✏ > 0, there exists an N such that for all n > N , |an L| <= ✏.
A sequence {an }+1
= a0 , a1 , a2 , . . . is = increasing if
0
an  an+1 for all n 0.
A sequence {an }+1
= a0 , a1 , a2 , . . . is = strictly increasing if
0
an < an+1 for all n 0.
A sequence {an }+1
= a0 , a1 , a2 , . . . is = decreasing if
0
an an+1 for all n 0.
A sequence {an }+1
= a0 , a1 , a2 , . . . is = strictly decreasing if
0
an > an+1 for all n 0.
A sequence {an }+1
= a0 , a1 , a2 , . . . is = monotonic if
0
it is either increasing or decreasing.
A series is writeen this way:
+1
X
n=0
an , which means a0 + a1 + a2 + = · · ·
a0 , a1 , a2 , . . . is called the sequence of underlying = terms.
There are two sequences associated with every series:
The sequence {sn }+1
= s0 , s1 , s2 , . . . , = where
0
s 0 = a0
s 1 = a0 + a1
s 2 = a0 + a1 + a2
..
.
s n = a0 + a1 + a2 + · · · + an
sn+1 = a0 + a1 + a2 + · · · + an = +an+1
is the sequence of partial sums.
A series converges if and only if its sequence of partial sums =
converges.
sequences, . . . , sequences, . . .
1. Find the general term for each of the following sequences.
Then determine whether or not the sequence converges. If the sequence does converge, find its limit.
(a)
1 1
1
1
1
,
,
, =
,
,...
4 16 64
256 1024
(b)
1 2 3
4 5
, , , = , ,...
3 4 5
6 7
(c)
1,
1 1
, ,
2 4
1
1
=,
,...
8
16
(d)
1, 1, 1, 1, 1, . . .