The Real And Complex Number Systems

n
c n ∑ a k b n−k .
k0
Proof: ByTheorem 8.44 and Theorem 8.13, it is clear.
Remark: We can use Mertens’ Theorem, thenitisclear.
8.34 Aseriesoftheform∑
n1
a n /n s is called a Dirichlet series. Given two absolutely
convergent Dirichlet series, say ∑
n1
a n /n s and ∑
n1
b n /n s , having sums As and Bs,
respectively, show that ∑ n1
c n /n s AsBs, wherec n ∑ d|n
a d b n/d .
Proof: ByTheorem 8.44 and Theorem 8.13, wehave
∑
n1
a n /n s
∑
n1
b n /n s ∑
n1
where
C n ∑ a d d −s b n/d n/d −s
d|n
n −s ∑ a d b n/d
d|n
c n /n s .
So,wehaveprovedit.
8.35 s ∑
n1
1/n s , s 1, show that 2 s ∑
n1
dn/n s ,wheredn is the
number of positive divisors of n (including 1 and n).
Proof: ItisclearbyExercise 8.34. So, we omit the proof.
Ces’aro summability
8.36 Show that each of the following series has C,1 sum 0 :
(a) 1 − 1 − 1 1 1 − 1 − 1 1 1 −−.
Proof: It is clear that |s 1 ...s n| ≤ 1 for all n, wheres n means that the nth partial sum
of given series. So,
s 1 ...s n
n ≤ 1 n
which implies that the given series has C,1 sum 0.
(b) 1 − 1 1 1 − 1 1 1 − 1 − .
2 2 2 2 2
Proof: It is clear that |s 1 ...s n| ≤ 1 for all n, wheres 2
n means that the nth partial
sum of given series. So,
s 1 ...s n
n ≤
2n
1
which implies that the given series has C,1 sum 0.
(c) cos x cos3x cos5x (x real, x ≠ m).
Proof: Lets n cosx ... cos2n − 1x, then
C n