Dirichlet's Test for Uniform Convergence

Theorem

Suppose:

• The partial sums of $\displaystyle \sum_{n=1}^\infty a_n (x)$ are bounded on $D$.

• ${b_n(x)}$ is monotonic for each $x\in D$.

• $b_n(x)\to 0$ converges uniformly on $D$.

Then:

$\displaystyle \sum_{n=1}^{\infty}a_n(x)b_n(x)$ converges uniformly on $D$.

Proof

Suppose $b_n(x)\geq b_{n+1}(x)$ for each $x \in D$.

All we need to show is that $\displaystyle \sum_{n=1}^{\infty}|b_n(x)-b_{n+1}(x)|$ converges uniformly on $D$.

To do this we show that the Cauchy Criterion holds.

Assign $\epsilon<0$, then $\exists N \in \N$ such that $\displaystyle \forall x \in D, \forall n \ge N: \left \vert {b_n(x)} \right \vert < \frac \epsilon 2$.

If $x\in D$ and $n > m \ge N$ then,

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \sum_{k=m+1}^n \left \vert {b_k(x)-b_{k+1}(x)} \right \vert\)	\(=\)	\(\displaystyle \)	\(\displaystyle \sum_{k=m+1}^n(b_k(x)-b_{k+1}(x))\)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \)	\(\displaystyle b_{m+1}(x)-b_{n+1}(x)\)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \)	\(\displaystyle \left \vert {b_{m+1}(x)-b_{n+1}(x)} \right \vert\)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\le\)	\(\displaystyle \)	\(\displaystyle \left \vert {b_{m+1}(x)+b_{n+1}(x)} \right \vert\)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(<\)	\(\displaystyle \)	\(\displaystyle \frac \epsilon 2 + \frac \epsilon 2\)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \)	\(\displaystyle \epsilon\)	\(\displaystyle \)	\(\displaystyle \)

$\blacksquare$

Source of Name

This entry was named for Johann Lejeune Dirichlet.

Sources

• Bruce Watson: Real Analysis Course Notes, Memorial University (2007)