In mathematical analysis, Tannery's theorem gives sufficient conditions for the
interchanging of the limit and infinite summation operations. It is named after
Jules Tannery
Jules Tannery (24 March 1848 – 11 December 1910) was a French mathematician, who notably studied under Charles Hermite and was the PhD advisor of Jacques Hadamard. Tannery's theorem on interchange of limits and series is named after him. He ...
.
Statement
Let
and suppose that
. If
and
, then
.
Proofs
Tannery's theorem follows directly from Lebesgue's
dominated convergence theorem
In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the ''L''1 norm. Its power and utility are two of the primary ...
applied to the
sequence space
In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural num ...
.
An elementary proof can also be given.
Example
Tannery's theorem can be used to prove that the binomial limit and the infinite series
characterizations of the exponential are equivalent. Note that
:
Define
. We have that
and that
, so Tannery's theorem can be applied and
:
References
{{Reflist
External links
Generalizations of Tannery's Theorem Mathematical analysis
Limits (mathematics)