Kenneth Kuttler

6.6. DOUBLE SERIES 165

Proof: Note that for all a,b, f (a,b) ≤ supb∈B supa∈A f (a,b) and therefore, for all a,supb∈B f (a,b)≤ supb∈B supa∈A f (a,b). Therefore,

supa∈A

supb∈B

f (a,b)≤ supb∈B

supa∈A

f (a,b) .

Repeat the same argument interchanging a and b, to get the conclusion of the lemma.

Theorem 6.6.4 Let ai j ≥ 0. Then ∑∞i=1 ∑

∞j=1 ai j = ∑

∞j=1 ∑

∞i=1 ai j.

Proof: First note there is no trouble in defining these sums because the ai j are allnonnegative. If a sum diverges, it only diverges to ∞ and so ∞ is the value of the sum. Nextnote that ∑

∞j=r ∑

∞i=r ai j ≥ supn ∑

∞j=r ∑

ni=r ai j because for all j,∑∞

i=r ai j ≥ ∑ni=r ai j.Therefore,

using Lemma 6.1.3,

∞

∑j=r

∞

∑i=r

ai j ≥ supn

∞

∑j=r

∑i=r

ai j = supn

limm→∞

∑j=r

∑i=r

ai j = supn

limm→∞

∑i=r

∑j=r

ai j

= supn

∑i=r

limm→∞

∑j=r

ai j = supn

∑i=r

∞

∑j=r

ai j = limn→∞

∑i=r

∞

∑j=r

ai j =∞

∑i=r

∞

∑j=r

ai j

Interchanging the i and j in the above argument proves the theorem.The following is the fundamental result on double sums.

Theorem 6.6.5 Let ai j ∈ R and suppose ∑∞i=r ∑

∞j=r∣∣ai j∣∣< ∞ . Then ∑

∞i=r ∑

∞j=r ai j =

∑∞j=r ∑

∞i=r ai j and every infinite sum encountered in the above equation converges.

Proof: By Theorem 6.6.4, ∑∞j=r ∑

∞i=r∣∣ai j∣∣ = ∑

∞i=r ∑

∞j=r∣∣ai j∣∣ < ∞. Therefore, for each

j, ∑∞i=r∣∣ai j∣∣ < ∞ and for each i, ∑

∞j=r∣∣ai j∣∣ < ∞. By Theorem 6.2.2 on Page 156, both of

the series ∑∞i=r ai j, ∑

∞j=r ai j converge, the first one for every j and the second for every i.

Also, ∑∞j=r∣∣∑∞

i=r ai j∣∣≤ ∑

∞j=r ∑

∞i=r∣∣ai j∣∣< ∞ and ∑

∞i=r∣∣∑∞

j=r ai j∣∣≤ ∑

∞i=r ∑

∞j=r∣∣ai j∣∣< ∞ so by

Theorem 6.2.2 again, ∑∞j=r ∑

∞i=r ai j, ∑

∞i=r ∑

∞j=r ai j both exist. It only remains to verify they

are equal.By Theorem 6.6.4 and Theorem 6.1.6 on Page 154

∞

∑j=r

∞

∑i=r

∣∣ai j∣∣+ ∞

∑j=r

∞

∑i=r

ai j =∞

∑j=r

∞

∑i=r

(∣∣ai j∣∣+ai j

∞

∑i=r

∞

∑j=r

(∣∣ai j∣∣+ai j

∞

∑i=r

∞

∑j=r

∣∣ai j∣∣+ ∞

∑i=r

∞

∑j=r

ai j =∞

∑j=r

∞

∑i=r

∣∣ai j∣∣+ ∞

∑i=r

∞

∑j=r

ai j

and so ∑∞j=r ∑

∞i=r ai j = ∑

∞i=r ∑

∞j=r ai j.It follows the two series are equal.

One of the most important applications of this theorem is to the problem of multiplica-tion of series.

Definition 6.6.6 Let ∑∞i=r ai and ∑

∞i=r bi be two series. For n ≥ r, define

cn ≡n

∑k=r

akbn−k+r.

The series ∑∞n=r cn is called the Cauchy product of the two series.

6.6. DOUBLE SERIES 165Proof: Note that for all a,b, f (a,b) < suppep supzc, f (a,b) and therefore, for all a,supyeg Sf (a,b) < suppeg SUpgc, f (a,b). Therefore,sup sup f (a,b) < supsup f (a,b).acA beB beBacARepeat the same argument interchanging a and b, to get the conclusion of the lemma. §fTheorem 6.6.4 Let aj; > 0. Then V2, Y?-) ij = LF LE az.Proof: First note there is no trouble in defining these sums because the q;; are allnonnegative. If a sum diverges, it only diverges to co and so © is the value of the sum. Nextnote that )5"_. Vj-,aij = sup, Lj, Li, aij because for all j, "aij > Li, aij-Therefore,using Lemma 6.1.3,co on nmY Lay >supY Yay =sup lim LY Lay = sup lim ay Gij= =ri=r n j= =ri=r * j= ri=r i= rj= r=sp lim bey = sup) Yai = = lim Yay =Y Yaype i=r j=r nveois rj=r i=r j=rInterchanging the i and j in the above argument proves the theorem. JJThe following is the fundamental result on double sums.Theorem 6.6.5 Let a;; ¢ R and suppose Y2_,.Y_,.|ai;| <0 . Then L2_.L 2, aij =Yij-r Li-r Gij and every infinite sum encountered in the above equation converges.Proof: By Theorem 6.6.4, Y5~,. Lin, lai; = Li Lis |ai;| < oo, Therefore, for eachI, Ley |ai| < eo and for each i, Yi, |ai)| < co. By Theorem 6.2.2 on Page 156, both ofthe series )/j-.aij, Lj—,4ij converge, the first one for every j and the second for every i.Also, Yi=r ye, aij| < Lier Yi-r i=r YF, aij| < Lie, Yi=r |ai| < © SO byTheorem 6.2.2 again, )7-_,. Li, 4ij, Li, Lj, aij both exist. It only remains to verify theyare equal.By Theorem 6.6.4 and Theorem 6.1.6 on Page 154yy |aij| + yay = yy ({ai;| + ais)j=ri=r j=ri=r j=ri=r= Dd (lala) =e LV fail +L eau = Le Vela + Le Lasi=r j=r i=r j=r i=r j=r j=ri=r i=r j=rand so )¥7_,. Li, ij = Li=r Lj=r 4ij-It follows the two series are equal.One of the most important applications of this theorem is to the problem of multiplica-tion of series.Definition 6.6.6 Le Yi, ai and Y-,b; be two series. For n > r, definenCn = y axDn—k+r-k=rThe series )\y_,.Cn is called the Cauchy product of the two series.