Graphics Reference
In-Depth Information
1
n
dx
x
1
n
?
Why is
1
Provethat
1
r
dx
x
1
r
.
1
r
. Provethat
1
n
s
Let
s
log
n
1.
Let
D
s
log
n
. Calculatesomevalus of
D
, using a computer
if possible.
dx
x
1
n
1
.
Show that
D
D
Deduce that (
D
) is a monotonic decreasing sequence of positive
terms and so has a limit. The limit, in this case, is usually denoted
by
, and is known as Euler's constant. Its value is slightly greater
than 0.577.
Let
E
s
log
(
n
1). Check that (
E
) is an increasing sequence,
that (
D
E
) is a null sequence (so that (
D
) and (
E
) havethe
same limit), and illustrate an area equal to
E
on a graph.
f
(1)
f
(2)
f
(3)
1
2
3
4
5
6
57
The integral test
Let
f
(
x
)
0, where
f
is a decreasing function for
x
1. What may
be said about the sequence (
f
(
n
))? Is it necessarily convergent? Since
f
is monotonic,
f
is integrable, see qns 10.7 and 10.8. Why is
f
(
n
1)
f
(
x
)
dx
f
(
n
)?
Deduce that
f
(
x
)
dx
f
(
r
)
f
(
r
).
