Category:Dirichlet's Test for Uniform Convergence

This category contains pages concerning Dirichlet's Test for Uniform Convergence:

Let $D$ be a set.

Let $\struct {V, \norm {\,\cdot\,} }$ be a normed vector space.

Let $a_i, b_i$ be mappings from $D \to M$.

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Let the following conditions be satisfied:

$(1): \quad$ The sequence of partial sums of $\ds \sum_{n \mathop = 1}^\infty \map {a_n} x$ be bounded on $D$

$(2): \quad \sequence {\map {b_n} x}$ be monotonic for each $x \in D$

This article, or a section of it, needs explaining. In particular: But $V$ is not ordered. Need to assume $V=\R$? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

$(3): \quad \map {b_n} x \to 0$ converge uniformly on $D$.

Then:

$\ds \sum_{n \mathop = 1}^\infty \map {a_n} x \map {b_n} x$ converges uniformly on $D$.