Dirichlet's Test for Uniform Convergence

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Theorem

Suppose:




Then:

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


Proof

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

All we need to show is that $\displaystyle \sum_{n \mathop = 1}^\infty \left\vert{b_n(x)-b_{n+1}(x)}\right\vert$ 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,

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

$\blacksquare$


Source of Name

This entry was named for Johann Lejeune Dirichlet.


Sources

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