Graphics Reference
In-Depth Information
Limit form of second comparison test
qn 55
If 0
a
,0
b
and (
a
/
b
)
l
0, then
a
and
b
are both convergent or both divergent.
Integral test
qn 57
If, for
x
1,
f
(
x
) is a decreasing positive
function, then
f
(
n
) is convergent if and only if
f
(
x
)
dx
exists.
Series with positive and negative terms
Alternating series test
S
n
1
0.9
0.8
0.7
0.6
0.5
n
5
10
15
20
62 From our knowledge of
1/
n
we might expect the series
1
2
1
3
1
4
1
5
(
1)
1
...
...
n
to be divergent.
However, if we let the
n
th partial sum
s
(
1)
/
r
, and
examine
s
,
s
s
,
s
s
, we can see that all these are positive.
s
Provethat
s
is positive.
Deduce that the sequence (
s
) is strictly increasing.
Likewise, if we examine
s
s
,
s
s
, these are negative. Prove
that
s
is negative.
Deduce that the sequence (
s
s
) is strictly decreasing.
Theproposition
1
2
n
s
1
s
s
s
,
