Math 320, Real Analysis I Quiz 1 — Solutions 1. (a) Define what it ...

Math 320, Real Analysis I
Quiz 1 — Solutions
(c) Prove that every convergent sequence is a Cauchy sequence.
Proof. Let (a n ) be a convergent sequence with limit a = lim a n . Let ε > 0 be arbitrary.
Then ε/2 > 0 as well, so there is a natural number N ∈ N such that
|a n − a| < ε 2
whenever n ≥ N. Now suppose m, n ∈ N with m, n ≥ N and consider
|a m − a n | = |a m − a + a − a n | = |(a m − a) + (a − a n )| ≤ |a m − a| + |a − a n | < ε 2 + ε 2 = ε
by the Triangle Inequality. Therefore, |a m −a n | < ε whenever m, n ≥ N, so the sequence
(a n ) is Cauchy.
4. (a) Define what it means for an infinite series to converge.
Definition: We say that the infinite series ∑ b n converges to B if its sequence of partial
sums (s m ), whose m-th term is given by s m = b 1 + b 2 + b 3 + · · · + b m for all m ∈ N,
converges to B.
(b) Give an example of each of the following, or state that such a request is impossible by
referring to the proper theorem(s):
i. a convergent infinite series that is commutative.
Example: By the Rearrangement Theorem, every series that is absolutely convergent
is commutative. So the series
∞∑ 1
n 2 = 1 + 1 4 + 1 9 + 1 16 + · · · ,
n=1
which converges since it is a p-Series with p = 2 > 1 and is absolutely convergent
since ∑ ∣ ∣ ∣∣
1 ∣∣∣
n 2 = ∑ 1 again which converges, is an example of a convergent infinite
n2 series that is commutative.
ii. a convergent infinite series whose sequence of partial sums is unbounded.
Impossible! If ∑ b n is a convergent infinite series, then its sequence of partial sums,
(s m ), must converge by the definition of convergence of an infinite series. Yet, if (s m )
converges, then (s m ) is bounded, for every convergent sequence is bounded. Thus
it is impossible to find an example of a convergent infinite series whose sequence of
partial sums is unbounded.
iii. a convergent infinite series that is not absolutely convergent.
Example: The Alternating Harmonic Series,
∞∑ (−1) n+1
= 1 − 1 n 2 + 1 3 − 1 4 + · · · ,
n=1
is a convergent infinite series by the Alternating Series Test, but it is not absolutely
convergent since the associated series of absolute values, ∑ ∞
n=1 ∣ (−1)n+1
n
∣ =
∑ ∞
n=1 1 n = 1 + 1 2 + 1 3 + 1 4
+ · · · , is the Harmonic Series, which diverges.
iv. an absolutely convergent infinite series that is not convergent.
Impossible! By the Absolute Convergence Test, every absolutely convergent
series is convergent.