Tannery's Theorem

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.

Statement

Let $S_n = \sum_^\infty a_k(n)$ and suppose that $\lim_ a_k(n) = b_k$. If $, a_k(n), \le M_k$ and $\sum_^\infty M_k < \infty$, then $\lim_ S_n = \sum_^ b_k$.

Proofs

Tannery's theorem follows directly from Lebesgue's dominated convergence theorem applied to the sequence space $\ell^1$.

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 $e^x$ are equivalent. Note that

: $\lim_ \left(1 + \frac\right)^n = \lim_ \sum_^n \frac.$

Define $a_k(n) = \frac$. We have that $, a_k(n), \leq \frac$ and that $\sum_^\infty \frac = e^ < \infty$, so Tannery's theorem can be applied and

: $\lim_ \sum_^\infty \frac =\sum_^\infty \lim_ \frac =\sum_^\infty \frac = e^x.$

References

{{Reflist

External links

Generalizations of Tannery's Theorem
Mathematical analysis
Limits (mathematics)