Exercise.

A sequence (_{
)
} is given by

(a) Prove that

for all

(b) Deduce that _{
2
} for all

(c) Using induction, show that

for each

Exercise.

Suppose in each case that (_{
)
} is a sequence whose
_{
)
} converges and find the limit. If you use the
continuity of any standard functions in your answer, state which
function(s) you use that you know to be continuous and at what
value ^{2}

(a) ^{2+2+1
}
^{2+3}
^{2}
^{2}
^{2-(-1)2
}
^{2++1
}
^{2+1}

Exercise.

Prove that the following sequences (_{
)
} do not
converge, using the following method.

Suppose first that the sequence in question converges to
some ** _{
)
}**, (

The convergence or nonconvergence of any sequences
you use *must* be justified either by stating that they are "standard"
convergent/nonconvergent sequences discussed in lectures (such as
^{
}) or by giving a proof.

(a) (-1)^{
(1-2-)+1
}.

(b) (-1)^{
(2-)+
2
1+
}.

(c) ^{
1)
}

(d)

Exercise.

The sequence (_{
)
} is defined inductively
by _{1=}_{2=1} and
_{
+2=
}_{
+}_{
+1+1}
4
.
Prove the following by induction on

(a) _{
1
} for all

(b) _{
12
} for all

(d) Say whether (_{
)
} has a limit in

Exercise.

The sequence (_{
)
}
_{1=1}
_{2=12
}
_{
+2=3
+1+
4
}

(a) _{
1
}

(b) _{
+1-
=(-1)
24
}

(c) The subsequences (_{2-1)}
_{2)}
_{
+2-
}

(d) Explain why these results imply that (_{2-1)}
_{2)}
_{
)
}

(e) Find the limit _{
}

**Solution to 3.2(a).** Write

and observe that ^{2}
^{2}
^{2}

**Solution to 3.3(a).** Suppose _{
=(-1)
(1-2-)+1
}
defines a sequence converging to

Now 1 is a constant sequence, so converges to 1.
The sequence 2^{-
} converges to 0 by a
result from lectures. So
by the continuity of - at (0,1)
1-2^{-
1-0=1
} as _{
-1)
-1
} as

which is known from lectures to be nonconvergent.

**Solution to 3.4.** (a) Let

.
Then _{
1
}_{1=11
}.
Also _{2=11
}.
Now suppose

using _{
1
} and _{
+1
1
}.
So

(Note: If you are not happy about assuming
an induction hypothesis that says that

(1)

holds and(

(

(

. Often induction proofs require the induction hypothesis to cover all previous cases of the induction statement.)(1 +1 ( ))

(b) Let

. _{
12
}_{1=1
12
} and _{2=1
12
}. Now suppse

by

(c) Let the statement _{
}_{
+1
}_{
+1
}3
,
so as _{1=}_{2=1},

as _{
}_{
+1
}
and _{
+1
12
}. This proves

(d) These results show that (_{
)
} is bounded (by (a) and (b))
and is monotonic (by (c)), and so by monotone convergence
has a limit _{
+1)} and
(_{
+2)} are subsequences of (_{
)
}
so by the theorem on subsequences converge to the same limit _{
+2=
}_{
+}_{
+1+1}
4
,
so by the continuity of + and division by 4,
_{
+2
++1
4
}
as