Math 401
Sequences
Name:_________________________
Work the problems on a separate piece of paper and staple to this one and make sure your work is
neat and complete. You will likely need help on some of these problems, so please come and talk to
me if you get stuck.
Problem 1 Determine whether the sequence converges or diverges. If it converges, find the limit.
I
II
III
IV
√
Problem 2 Determine whether the sequence converges or diverges. If it converges, find the limit.
I
II
III
IV
Problem 3 Let
Problem 4 Let
of
{
( )
(
( ))
and use the squeezing theorem to show that
(
⟦ ⟧
) and
a real number such that
Then guess the value
and use the squeezing theorem show that your answer is correct.
Problem 5 Give an
proof that
(√
√ )
(Hint: use the conjugate).
Problem 6 Let {
}
{
} be two sequences such
Prove that
Problem 7 Let { } a sequence and
any subsequence of { } , then
Problem 8
bounded.
Let {
, where
.
} a sequence and
Prove that {
a real number such that
Problem 9 A sequence { } is defined by
√
{
}
to show that
is increasing and bounded above by 3. Find
Problem 10 A sequence {
Prove that if {
a real number such that
} is defined by
√
} is
} is
. Use mathematical induction
. Show that {
} is increasing
and bounded above by 3. Find
Problem 11 Let
and
√
. We then define
so that
is the arithmetic mean of
√
geometric mean of
. I Use mathematical induction to show that
use I and to show that
. III Deduce that {
convergent sequences and show that
.
and
and
}
{
is the
II Then
} are