c University of Bristol 2014
Further Topics in Analysis: Exercises 6
1. Let (an )n∈N be a sequence in (0, ∞). Set
l := lim sup a1/n
n .
n→∞
P∞
(i) Show that if 0 6 l < 1, then n=1 an converges.
P
(ii) Show that if l > 1, then ∞
n=1 an diverges.
(iii) Find λ ∈ [0, ∞) for which
∞
X
n
2n(−1) λn
n=1
converges respectively diverges.
Hint: For (i) and (ii), adapt the proof of the root test.
2. Let (an )n∈N+ and (bn )n∈N+ be bounded sequences in R.
(a) Show that
lim inf { an − bn } > lim inf an − lim sup bn .
n→∞
n→∞
n→∞
It may be helpful to use the formula given in lectures.
(b) Formulate a counterpart to this result for the limit superior.
3. Let (an )n∈N+ and (bn )n∈N+ be bounded sequences in [0, +∞).
(a) Show that
lim sup an bn 6 lim sup an · lim sup bn .
n→∞
n→∞
n→∞
(b) Demonstrate that the inequality in (a) need not hold for bounded sequences in R.
4. Describe the set of all accumulation points and find the limit superior and limit inferior
of the following sequences.
√
√
(a) an = n + 1 + (−1)n n − 1;
√
(b) an =
(c) an =
√
n+1+(−1)n n−1
;
n
√
√
n+1+(−1)n n−1
√
;
n
(d) (an )n∈N = (0, 1, 0, −1, 0, 1, 0, −1, 0, 1, 0, −1, . . . ).
5. Let (an )n∈N be a bounded sequence of real numbers. Which of the following must be
true? Give counterexamples for those that may be false.
(a) lim supn→∞ an = lim inf n→∞ an .
(b) (an )n∈N has a convergent subsequence tending to lim supn→∞ an .
(c) If (bm )m∈N is a convergent subsequence of (an )n∈N , then either
bm → lim sup an
n→∞
or bm → lim inf an
n→∞
as m → ∞.
(d) If (bm )m∈N is a convergent subsequence of (an )n∈N , with limm→∞ bm = b then
lim inf an 6 b 6 lim sup an .
n→∞
n→∞
(e) For every real number b with
lim inf an 6 b 6 lim sup an ,
n→∞
n→∞
there is a convergent subsequence (bm )m∈N of (an )n∈N such that limm→∞ bm = b.
6. Determine whether or not the following sequences (an )n∈N are Cauchy sequences, and
justify your answer directly from the definition.
(a) an =
n
n+1 ;
(b) an =
2n +1
2n ;
(−1)n (2
(c) an =
+
1
).
n2
7. (a) Let (an )n∈N be a sequence such that
(∀n ∈ N)(|an+1 − an | 6
1
).
2n
Prove that (an )n∈N is a Cauchy sequence.
(b) Let (an )n∈N be the sequence defined by a1 = 1, a2 = 1/2 and
1
an+2 = (an+1 + an ) for all n > 1.
2
Prove that (an )n∈N converges.
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