Exercise.
For each of the following, decide (with justification) which of the
following series converge.
(f)
=2
(
2+-1)
12
(
5-2)
12
(g)
=2
(
2+-1)
12
(
3-2)
12
(h)
=1
2
(Use the ratio test.)
(i)
=1
(You may find it helpful to
recall that (1+1)
=2.71828...
as
.)
Exercise.
(a) Prove that
=1
diverges for all
0
. (Use the null sequence test.)
(b) Prove that
=1
2+1
diverges for all
1
and converges for 0
1
.
(Use the ratio test.)
(c) Prove that
=1
converges for all
0
.
Exercise.
(a)
=1
+
+
converges when 0
1
and diverges when
1
.
(b)
+2(+1)2
2
+3(+2)3
3
+ converges when 0
and diverges when
.
Exercise.
Discuss the convergence of
12
+
1325
2
+
135
258
3
+
1357
25811
4
+
Solution to selected parts of these exercises
Exercise 5.1(a)
The series has positive terms for
2
and
+1
3-2
2
3/2
=
4
2
for
2
, so
=2
+1
3-2
converges by comparison test with
=2
1
2
hence
=1
+1
3-2
also
converges. (OR: you could use the limit comparison test here.)
Exercise 5.1(b)
The series has positive terms for
2
and
-1
2-2
/22
2
=
14
1
for
2
, so
=2
-1
2-2
diverges by comparison test with
=2
1
hence
=1
-1
2-2
also
diverges. (OR: you could use the limit comparison test here.)
Exercise 5.3(a)
For 1
0
, the terms in the series
=1
+
+
are all positive. Let
=
+
+
.
We apply the ratio test.
+1
=
++1
++1
+1
+
+
-
=
(
+1+1
)(
+1+1
)
(
+1)(
+1)
which converges to by continuity of the arithmetic operations and
1
0. Therefore by the ratio test, the series converges if
1
and diverges if
1
. If =1 the series is
=1
+
+
and
+
+
=
/+1
/+1
1
so the series diverges by the null sequence test.