multiplication of series
Theorem (Franz Mertens). If the series and with real or complex converge and have the sums (http://planetmath.org/SumOfSeries) and , respectively, and at least one of them converges absolutely, then also the series
is convergent and its is equal to .
Proof. Denote the partial sums of the series , and for each . Then we have and . Suppose that e.g. the series converges absolutely and that at least one is distinct from zero; so the is a real positive number . Let be an arbitrary positive number.
Now we can write the identities
There is a positive number such that when . Then
The convergence of implies that there is a number such that when . Thus we have
if . Because , the numbers are bounded, i.e. there is a positive number such that for each we have and consequently . It follows that for every . We apply Cauchy criterion for convergence to the series getting a number such that for each , one has the inequality if . Accordingly we obtain the estimation
which is valid when .
If we choose and such that , then the inequalities (2), (3) and (4) are satisfied, ensuring that
This means that the assertion of the theorem has been proved.
Remark. The mere convergence of both series does not suffice for convergence of (1). This is seen in the following example by Cauchy where both series are
They converge by virtue of Leibniz test, but not absolutely (see the -test (http://planetmath.org/PTest)). In their product series
the absolute value of the is , having summands which all are greater than (this is seen when one looks at the half circle or , which shows that and thus ). Because , the product series does not satisfy the necessary condition of convergence (http://planetmath.org/ThenA_kto0IfSum_k1inftyA_kConverges) and therefore the series diverges.
- 1 Ernst Lindelöf: Johdatus funktioteoriaan. Mercatorin Kirjapaino Osakeyhtiö. Helsinki (1936).
|Title||multiplication of series|
|Date of creation||2014-10-31 20:23:29|
|Last modified on||2014-10-31 20:23:29|
|Last modified by||pahio (2872)|
|Synonym||Cauchy multiplication rule|