The Real And Complex Number Systems

Also, lim q→ fp, q lim q→ p sin 1 2p
lim p→ lim q→ fp, q 0.
8.29 Prove the following statements:
sin q 2p
sin
q1
2p
q 2
0since|sin x| ≤ 1. So,
(a) A double series of positive terms converges if, and only if, the set of partial sums is
bounded.
Proof: ()Suppose that ∑ m,n
fm, n converges, say ∑ m,n
fm, n A 1 , then it means
that lim p,q→ sp, q A 1 . Hence, given 1, there exists a positive integer N such that as
p, q ≥ N, wehave
|sp, q| ≤ |A 1 | 1.
So, let A 2 maxsp, q :1≤ p, q N, wehave|sp, q| ≤ maxA 1 , A 2 for all p, q.
Hence, we have proved the set of partial sums is bounded.
()Suppose that the set of partial sums is bounded by M, i.e., if
S sp, q : p, q ∈ N, then sup S : A ≤ M. Hence, given 0, then there exists a
sp 1 , q 1 ∈ S such that
A − sp 1 , q 1 ≤ A.
Choose N maxp 1 , q 1 , then
A − sp, q ≤ A for all p, q ≥ N
since every term is positive. Hence, we have proved lim p,q→ sp, q A. Thatis,
∑ m,n
fm, n converges.
(b) A double series converges if it converges absolutely.
p
Proof: Lets 1 p, q ∑ m1
q
∑ n1
p
|fm, n| and s 2 p, q ∑ m1
q
∑ n1
fm, n, wewant
to show that the existence of lim p,q→ s 2 p, q by the existence of lim p,q→ s 1 p, q as
follows.
Since lim p,q→ s 1 p, q exists, say its limit a. Then lim p→ s 1 p, p a. It implies that
lim p→ s 2 p, p converges, say its limit b. So, given 0, there exists a positive integer N
such that as p, q ≥ N
|s 1 p, p − s 1 q, q| /2
and
|s 2 N, N − b| /2.
So, as p ≥ q ≥ N,
|s 2 p, q − b| |s 2 N, N − b s 2 p, q − s 2 N, N|
/2 |s 2 p, q − s 2 N, N|
/2 s 1 p, p − s 1 N, N
/2 /2
.
Similarly for q ≥ p ≥ N. Hence, we have shown that
lim
p,q→ s 2p, q b.
That is, we have prove that a double series converges if it converges absolutely.
(c) ∑ m,n
e −m2 n 2
converges.
Proof: Letfm, n e −m2 n 2
, then by Theorem 8.44, we have proved that