Graphics Reference
In-Depth Information
a
b
Theorem
qn 111
If
are both absolutely convergent,
then their Cauchy product is convergent to the
product of their sums.
and
Historical note
Archimedes (250 B.C.) could sum the series
(
)
. Oresme
(1323
—
1382) was ableto sum
)
and healso showed the
divergence of the harmonic series in the way we have done.
P. Mengoli found
(
1)
/
n
log 2 and summed the series
(
)
and
n
(
1/
n
(
n
1) in 1650. It was themthod of summing this latter series
which formed the basis of Leibniz' theory of integration. I. Newton in
1667 and N. Mercator in 1668 obtained the result
(
1)
x
log(1
x
) by integrating the power series for (1
x
)
.
James Gregory in 1671 and Leibniz in 1673 obtained the result
(
1)
/(2
n
1)
by integrating the power series for (1
x
)
.
During the seventeenth and eighteenth centuries the word 'series' was
equally used to describe a sequence or a series, and the word
'convergence' was used to refer to the sequence (
a
) or the series
a
,
even if it was the latter which appeared to beundr discussion. In the
eighteenth century, discussion of the sums of series was not confined to
those cases where the sequence of partial sums was convergent.
Insistence on at least the boundedness of partial sums for meaningful
discussion of an overall sum is due to Gauss (1813). In 1742 C.
Maclaurin used integrals to approximate to values of series and vice
versa, and illustrated his method with
. In 1748, by factorising the
power series for sin
x
like a polynomial, L. Euler showed that
1/
n
/6. In 1785 E. Waring showed that
1/
n
was
convergent/divergent according as
/
1. His proof used a
rudimentary form of the integral test. The proof we have given is a
special form of Cauchy's condensation test (1821): that
a
1/
n
is
convergent if and only if
(
k
)
a
is convergent when (
a
) is a decreasing
sequence of positive terms.
The title of d'Alembert's ratio test derives from work which he
published in 1768 establishing the convergence of the binomial series by
comparison with geometric progressions with common ratio less than 1.
The ratio test was used with modern precision in Gauss' paper on the
hypergeometric series (1813). In 1815, Fourier proved that e, defined by
the series
1/
n
!, was irrational. Thefirst comparison tst, or sandwich
theorem, is described in relation to geometric progressions by Bolzano
(1816). Bolzano (1817) also explained why, for
a
to be convergent, it
is necessary for (
a
) to be a null sequence, but not suMcient because of
