Graphics Reference
In-Depth Information
Answers and comments
1
a
c
can be extended to
finite
sums. If thesum is
infinite,
S
may have different values depending on how brackets are
inserted. So an agreed system for inserting brackets is needed.
(
b
c
)
(
a
b
)
3
1/(
r
(
r
1))
1/
r
1/(
r
1).
1
n
1
n
1
s
(1
)
(
)
...
.
1
1
1)
1
1
.
r
(
r
n
5
(1
2.
You do not 'get there' in a finite number of steps, but you can get as
closeas you lik.
(
)
)/(1
). Using 3.39 on geometric progressions, 1/(1
)
6
0.111 . . ..
7
9 ยท (10
)
0.090 909 . . ..
8
10.
9
(i) Convergent, from 3.39 and 3.42.
(ii) Divergent from 3.21. Partial sums tend to infinity when
x
1.
Partial sums
(1
(
1)
) when
x
1.
10
s
(
n
1)/2
2
(
n
3)/2
s
gives the inductive step.
Put
k
1 and
x
in 3.74(i).
11 lim
a
s
s
0.
13
0
(
n
)
n
is divergent.
14 If
s
is thepartial sum of thefirst
n
terms of
a
a
a
...
a
...
then
s
a
is thesum of thefirst
n
1 terms of
a
a
a
...
a
...
.
s
lim(
s
a
)
s
a
a
a
s
a
.
t
s
a
.
Conversely
a
t
a
t
a
.