Tannery's Theorem

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Theorem

Let $\sequence {p_n}$ be an increasing, unbounded above sequence of natural numbers.

For every $r, n \in \N$, let $v_r$ be a mapping:

$\map {v_r} n \in \C$

Let $\sequence {w_r}$ be a sequence of complex numbers such that, for every $r \in \N$:

$\ds w_r = \lim_{n \mathop \to \infty} \map {v_r} n$

Let $\sequence {M_r}$ be a sequence of non-negative real numbers such that:

$\forall n \in \N, r \le p_n: \size {\map {v_r} n} \le M_r$

and:

$\ds \sum_{r \mathop = 0}^\infty M_r$ is convergent

Then:

$\ds \lim_{n \mathop \to \infty} \sum_{r \mathop = 0}^{p_n} \map {v_r} n = \sum_{r \mathop = 0}^\infty w_r$

Proof

First, observe that for all $r \in \N$:

$\ds \size {w_r}$

$=$

$\ds \size {\lim_{n \mathop \to \infty} \map {v_r} n}$

$\ds $

$=$

$\ds \lim_{n \mathop \to \infty} \size {\map {v_r} n}$

Complex Modulus Function is Continuous

$\ds $

$\le$

$\ds M_r$

Lower and Upper Bounds for Sequences $(2)$

Let $\epsilon > 0$ be arbitrary.

By Tail of Convergent Series tends to Zero, select some $q$ such that:

$\ds \sum_{r \mathop = q}^\infty M_r < \frac \epsilon 3$

As $\sequence {p_n}$ is unbounded above, there is some $M \in \N$ such that:

$p_M \ge q$

Since $\sequence {p_n}$ is increasing:

$\forall n \ge M: p_n \ge p_M \ge q$

Additionally:

$\ds \lim_{n \mathop \to \infty} \paren {\sum_{r \mathop = 0}^{q - 1} \map {v_r} n - \sum_{r \mathop = 0}^{q - 1} w_r}$

$=$

$\ds \lim_{n \mathop \to \infty} \sum_{r \mathop = 0}^{q - 1} \paren {\map {v_r} n - w_r}$

$\ds $

$=$

$\ds \sum_{r \mathop = 0}^{q - 1} \lim_{n \mathop \to \infty} \paren {\map {v_r} n - w_r}$

Sum Rule for Complex Sequences

$\ds $

$=$

$\ds \sum_{r \mathop = 0}^{q - 1} 0$

$\ds \lim_{n \to \infty} \map {v_r} n = w_r$

$\ds $

$=$

$\ds 0$

Therefore, we can find some $N \in \N$ such that:

$\ds \size {\sum_{r \mathop = 0}^{q - 1} \map {v_r} N - \sum_{r \mathop = 0}^{q - 1} w_r} < \frac \epsilon 3$

Let $n \ge \max \set {M, N}$ be arbitrary.

Then:

$\ds \size {\sum_{r \mathop = 0}^{p_n} \map {v_r} n - \sum_{r \mathop = 0}^\infty w_r}$

$=$

$\ds \size {\sum_{r \mathop = 0}^{q - 1} \map {v_r} n - \sum_{r \mathop = 0}^{q - 1} w_r + \sum_{r \mathop = q}^{p_n} \map {v_r} n - \sum_{r \mathop = q}^\infty w_r}$

$n \ge M \implies p_n \ge q$

$\ds $

$\le$

$\ds \size {\sum_{r \mathop = 0}^{q - 1} \map {v_r} n - \sum_{r \mathop = 0}^{q - 1} w_r} + \size {\sum_{r \mathop = q}^{p_n} \map {v_r} n} + \size {\sum_{r \mathop = q}^\infty w_r}$

General Triangle Inequality for Complex Numbers

$\ds $

$\le$

$\ds \size {\sum_{r \mathop = 0}^{q - 1} \map {v_r} n - \sum_{r \mathop = 0}^{q - 1} w_r} + \sum_{r \mathop = q}^{p_n} \size {\map {v_r} n} + \sum_{r \mathop = q}^\infty \size {w_r}$

Triangle Inequality for Indexed Summations and Triangle Inequality for Series

$\ds $

$\le$

$\ds \size {\sum_{r \mathop = 0}^{q - 1} \map {v_r} n - \sum_{r \mathop = 0}^{q - 1} w_r} + \sum_{r \mathop = q}^\infty M_r + \sum_{r \mathop = q}^\infty M_r$

$\size {\map {v_r} n} \le M_r$ and $\size {w_r} \le M_r$

$\ds $

$<$

$\ds \frac \epsilon 3 + \frac \epsilon 3 + \frac \epsilon 3$

$\ds $

$=$

$\ds \epsilon$

Thus, for any $\epsilon > 0$, we have:

$\ds \forall n \ge \max \set {M, N}: \size {\sum_{r \mathop = 0}^{p_n} \map {v_r} n - \sum_{r \mathop = 0}^\infty w_r} < \epsilon$

Thus, by definition of a limit:

$\ds \lim_{n \to \infty} \sum_{r \mathop = 0}^{p_n} \map {v_r} n = \sum_{r \mathop = 0}^\infty w_r$

$\blacksquare$

Source of Name

This entry was named for Jules Tannery.

Sources

1926: T.J.I'A. Bromwich: An Introduction to the Theory of Infinite Series (2nd ed.): Chapter $\text {VII}$: Article $49$