Graphics Reference
In-Depth Information
The null sequence test
qn 11
If (
a
)is
not
a null sequence, then
a
is
divergent.
Convergence of geometric series
qns 2, 9
When
x
1,
x
1/(1
x
).
Series of positive terms
First comparison test
23 Let e
1
1
1/2!
1/3!
...
1/
n
!,
and let
s
, with
n
1
1
...
1/2
1.
Provethat e
3
1/2
.
Deduce that the sequence (e
s
) is monotonic increasing, bounded
above and so convergent. We say (e
)
e. [The relation between
this e and qn 4.36 is established in 11.28 and 11.32.]
24 Provethat eis irrational. Supposethat eis rational
p
/
q
, in lowest
terms. Show that e
e
k
/
q
! for some integer
k
. Check that
q
2,
and deduce that e
e
, contradicting the previous result
and showing that the hypothesis, of rational e, is false.
(1/
q
!) ·
25 If
a
0 for all
n
, prove that the sequence of partial sums of the
series
a
is monotonic increasing. Deduce that
a
is convergent
if and only if its partial sums arebounded.
26
The first comparison test
The result of qn 25 leads to a squeeze theorem for series of positive
terms.
If 0
a
b
, and
b
is convergent, show that the partial sums of
a
are bounded and deduce that
a
is convergent.
a
Contrapositively, if
is divergent, show that the partial sums of
b
areunbounded and so
b
is divergent.
27 Usethefact that
1
(
n
1)
1
n
(
n
1)
,
0
and qns 4 and 26 (thefirst comparison tst), to provethat
1/(
n
1)
is convergent.
Deduce from qn 14 that
1/
n
is convergent. (The sum of this
series is
/6. Po´ lya (1954), p. 19, gives a remarkable justification of
this dueto Eulr.)
