Exercise.

For each of the following, decide (with justification) which of the
following series converge.

(d)
=1
^{2+-1
}
^{4-3
2+1
}

(e)
=1
(+1)^{2-
2
}
^{3-
2+17
}

(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.