Dirichlet's Test for Uniform Convergence
Jump to navigation
Jump to search
Contents
Theorem
Let $D$ be a set.
Let $\struct {V, \norm {\,\cdot\,} }$ be a normed vector space.
Let $a_i, b_i$ be mappings from $D \to M$.
Let the following conditions be satisfied:
- $(1): \quad$ The sequence of partial sums of $\displaystyle \sum_{n \mathop = 1}^\infty \map {a_n} x$ be bounded on $D$
- $(2): \quad \sequence {\map {b_n} x}$ be monotonic for each $x \in D$
- $(3): \quad \map {b_n} x \to 0$ converge uniformly on $D$.
Then:
- $\displaystyle \sum_{n \mathop = 1}^\infty \map {a_n} x \, \map {b_n} x$ converges uniformly on $D$.
Proof
Suppose $\map {b_n} x \ge \map {b_{n + 1} } x$ for each $x \in D$.
All we need to show is that:
- $\displaystyle \sum_{n \mathop = 1}^\infty \size {\map {b_n} x - \map {b_{n + 1} } x}$
converges uniformly on $D$.
To do this we show that the Cauchy criterion holds.
Assign $\epsilon < 0$.
Then by definition of uniform convergence:
- $\displaystyle \exists N \in \N: \forall x \in D: \forall n \ge N: \size {\map {b_n} x} < \frac \epsilon 2$
Let $x \in D$ and $n > m \ge N$.
Then:
\(\displaystyle \sum_{k \mathop = m + 1}^n \size {\map {b_k} x - \map {b_{k + 1} } x}\) | \(=\) | \(\displaystyle \sum_{k \mathop = m + 1}^n \paren {\map {b_k} x - \map {b_{k + 1} } x}\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \map {b_{m + 1} } x - \map {b_{n + 1} } x\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \size {\map {b_{m + 1} } x - \map {b_{n + 1} } x}\) | |||||||||||
\(\displaystyle \) | \(\le\) | \(\displaystyle \size {\map {b_{m + 1} } x + \map {b_{n + 1} } x}\) | |||||||||||
\(\displaystyle \) | \(<\) | \(\displaystyle \frac \epsilon 2 + \frac \epsilon 2\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \epsilon\) |
$\blacksquare$
Source of Name
This entry was named for Johann Peter Gustav Lejeune Dirichlet.
Sources
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.28$: Dirichlet ($1805$ – $1859$)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: Dirichlet's test
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: Dirichlet's test