---
tag: Y0JH94QR80
kind: statement
role: theorem
name: Tannery's theorem
msc: [40A05, 40A30]
uses: [0CVVKY7GY2, 0AZ3R52593, 0W3K3QPEWT]
sources:
  - url: https://en.wikipedia.org/wiki/Tannery%27s_theorem
same_as:
  wikidata: Q60791768
  wikipedia: Tannery's theorem
---

## Statement

For each natural number $n$, let $(a_k(n))_{k \ge 0}$ be a sequence of complex numbers.
Assume that:

1. for each $k$, the limit $a_k = \lim_{n \to \infty} a_k(n)$ exists;
2. there are nonnegative real numbers $M_k$ with $\abs{a_k(n)} \le M_k$ for all $k$ and $n$, and $\sum_k M_k < \infty$.

Then the series $\sum_k a_k(n)$ and $\sum_k a_k$ [[converge absolutely|0W3K3QPEWT]], and
$$\lim_{n \to \infty} \sum_{k=0}^{\infty} a_k(n) = \sum_{k=0}^{\infty} a_k.$$

## Remarks

Given $\varepsilon > 0$, choose $K$ with $\sum_{k > K} M_k < \varepsilon$.
The tails beyond $K$ of both series are then smaller than $\varepsilon$, since $\abs{a_k} \le M_k$ too.
The finitely many terms with $k \le K$ converge, so the difference of the two sums is less than $3\varepsilon$ for large $n$.
Example: for real $x$ put $a_k(n) = \binom{n}{k} x^k / n^k$ for $k \le n$ and $a_k(n) = 0$ for $k > n$.
Then $\sum_k a_k(n) = (1 + x/n)^n$, $a_k(n) \to x^k/k!$, and $\abs{a_k(n)} \le \abs{x}^k / k!$.
So $(1 + x/n)^n \to \sum_k x^k / k! = e^x$.
Without the bound of clause 2 the conclusion fails: $a_k(n) = 1$ for $k = n$ and $0$ otherwise gives sums equal to $1$ but limits $a_k = 0$.

## Explanations

### L1 human

Tannery's theorem allows you to take a limit inside an infinite sum.
It needs one fixed summable bound for all the terms, uniformly in $n$.
Then no mass can escape to infinity, so the limit of the sums is the sum of the limits.

### L2 human

Tannery's theorem is the [[dominated convergence theorem|BYNV84PF86]] for the counting measure on $\N$, but it predates measure theory and has a short direct proof.
A typical use is to pass from finite products or binomial expansions to their limits, as in the proof that $(1 + x/n)^n \to e^x$.
Compare the [[Weierstrass M-test|0NX5NWAFC9]], which uses the same kind of bound to get uniform convergence of a series of functions.