1. If
is less than the radius of convergence of either of the series
,
, then the product of the two series is
, where
.
2. If the radius of convergence of is , and is the sum of the series when , and is less than either or unity, then , where .
3. Prove, by squaring the series for , that if .
4. Prove similarly that , the general term being .
5. The Binomial Theorem for a negative integral exponent. If , and is a positive integer, then
[Assume the truth of the theorem for all indices up to
. Then, by Ex. 2,
, where
as is easily proved by induction.]
6. Prove by multiplication of series that if and , then . [This equation forms the basis of Euler’s proof of the Binomial Theorem. The coefficient of in the product series is
This is a polynomial in and : but when and are positive integers this polynomial must reduce to in virtue of the Binomial Theorem for a positive integral exponent, and if two such polynomials are equal for all positive integral values of and then they must be equal identically.]
7. If then . [For the series for is absolutely convergent for all values of : and it is easy to see that if , , then .]
8. If then and
9. Failure of the Multiplication Theorem. That the theorem is not always true when and are not absolutely convergent may be seen by considering the case in which Then But , and so , which tends to ; so that is certainly not convergent.