## Limits of Sums, Differences, Products and Quotients of Series

Sums, products, differences and quotients of sequences converge in the obvious way.

Theorem

Ifconverges toandconverges tothenconverges to

Proof: ChooseThere is a positive integersuch that ifthenand a positive integersuch that ifthenLetthen ifand then

Henceconverges to

Theorem

Ifconverges toandconverges tothenconverges to

Proof:ChooseSincethere is a positive integersuch that ifthena positive integersuch that ifthenand a positive integersuch that ifthenLetthen ifand

so takethen

Theorem

Ifconverges toandconverges tothenconverges to

Proof: ChooseThere is a positive integersuch that ifthenand a positive integersuch that ifthenLetthen ifand then

Henceconverges to

Theorem

Ifconverges toandconverges tothenconverges to

Proof: ChooseSinceconverges, it is bounded so there existssuch thatfor allLetThere is a positive integersuch that ifthenand a positive integersuch that ifthenLetthen ifand then

Henceconverges to