56.1. RUNGE’S THEOREM 1769

56.1.3 Merten’s Theorem.

Theorem 56.1.5 Suppose ∑∞i=r ai and ∑

∞j=r b j both converge absolutely1. Then(

∞

∑i=r

ai

)(∞

∑j=r

b j

)=

∞

∑n=r

cn

where

cn =n

∑k=r

akbn−k+r.

Proof: Let pnk = 1 if r ≤ k ≤ n and pnk = 0 if k > n. Then

cn =∞

∑k=r

pnkakbn−k+r.

Also,

∞

∑k=r

∞

∑n=r

pnk |ak| |bn−k+r| =∞

∑k=r|ak|

∞

∑n=r

pnk |bn−k+r|

=∞

∑k=r|ak|

∞

∑n=k|bn−k+r|

=∞

∑k=r|ak|

∞

∑n=k

∣∣bn−(k−r)∣∣

=∞

∑k=r|ak|

∞

∑m=r|bm|< ∞.

Therefore,

∞

∑n=r

cn =∞

∑n=r

n

∑k=r

akbn−k+r =∞

∑n=r

∞

∑k=r

pnkakbn−k+r

=∞

∑k=r

ak

∞

∑n=r

pnkbn−k+r =∞

∑k=r

ak

∞

∑n=k

bn−k+r

=∞

∑k=r

ak

∞

∑m=r

bm

and this proves the theorem.It follows that ∑

∞n=r cn converges absolutely. Also, you can see by induction that you

can multiply any number of absolutely convergent series together and obtain a series whichis absolutely convergent. Next, here are some similar results related to Merten’s theorem.

1Actually, it is only necessary to assume one of the series converges and the other converges absolutely. Thisis known as Merten’s theorem and may be read in the 1974 book by Apostol listed in the bibliography.