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Newton's Principia (1687), to which Taylor applied a limiting process.
Maclaurin (1742) obtained his series by repeated differentiation of a
presumed infinite series expansion, putting x
0 at each stage
recognising it to be a special case of Taylor's. Lagrange (1797) having
'proved' that every function has a Taylor series expansion established
the First, Second and Third Mean Value Theorems as we have
described above. In each case he obtained bounds on the remainder
(really on K , in qns 13, 31 and 33) after one, two and three terms of the
Taylor series (respectively) and claimed, correctly, that his method
extended to n terms giving the Lagrange form of the remainder as we
know it today.
In 1823 Cauchy claimed that all the derivatives of exp(
1/ x
) are
zero at x 0 so that this function does not have a Taylor series at this
point. This undermined Lagrange's programme for a theory of analytic
functions based entirely on Taylor series. Cauchy derived his form of
the Taylor series remainder from the remainder in integral form, which
he also established in 1823.
In Weierstrass' lectures in 1861 he claimed the Fundamental
Theorem of Analysis to be that, when f ( x
) 0 for
i 1, 2, 3, . . ., n 1,
( x x
)
f ( x ) f ( x
)
ยท f ( x
( x x
)),
n !
for some ,0 1,
a form of Taylor's Theorem with Lagrange remainder which had been
obtained by Cauchy in 1829.
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