Limits of Sums, Differences, Products and Quotients of Series
Sums, products, differences and quotients of sequences converge in the obvious way.
Theorem
Ifconverges to
and
converges to
then
converges to
Proof: ChooseThere is a positive integer
such that if
then
and a positive integer
such that if
then
Let
then if
and
then
Henceconverges to
Theorem
Ifconverges to
and
converges to
then
converges to
Proof:Choose
Since
there is a positive integer
such that if
then
a positive integer
such that if
then
and a positive integer
such that if
then
Let
then if
and
so take
then
Theorem
Ifconverges to
and
converges to
then
converges to
Proof: ChooseThere is a positive integer
such that if
then
and a positive integer
such that if
then
Let
then if
and
then
Henceconverges to
Theorem
Ifconverges to
and
converges to
then
converges to
Proof: ChooseSince
converges, it is bounded so there exists
such that
for all
Let
There is a positive integer
such that if
then
and a positive integer
such that if
then
Let
then if
and
then
Henceconverges to