Winter 1995

Problem Statement: Let f_k:D\rightarrow\mathbb{R}, D\subseteq\mathbb{R} and let |f_k(x)|\leq M_k for all k\in\mathbb{N} and for every x\in D. Show that if \sum\limits_{k=1}^{\infty}M_k converges then \sum\limits_{k=1}^{\infty}f_k converges uniformly on D.

Note: This is asking us to prove the Weierstrass M-Test.

Proof: Let f_k:D\rightarrow\mathbb{R} and |f_k(x)|\leq M_k for every k\in\mathbb{N} and for every x\in D. Assume that \sum\limits_{k=1}^{\infty}M_k converges. This implies that the sequence of partial sums, \{T_K=\sum\limits_{k=1}^{K}M_k\}, converges. Every convergent sequence is Cauchy.

We wish to show that for every \varepsilon>0 there exists K\in\mathbb{N} such that for all n,m\geq K and for every x\in D it follows that |S_n(x)-S_m(x)|<\varepsilon. Note that S_N(x)=\sum\limits_{k=1}^{N}f_k(x).

Let \varepsilon>0. Then there exists a K\in\mathbb{N} such that for n,m\geq K it follows that |T_n-T_k|<\varepsilon. This implies that for n,m\geq K, |\sum\limits_{k=m}^{n}M_k|<\varepsilon.

Let n,m\geq K and without loss of generality suppose that n>m. Then it follows that:

|S_n(x)-S_m(x)|=|\sum\limits_{k=m}^{n}f_k(x)|

\leq |M_m+M_{m+1}+\dots+M_n| for every x\in D

=|\sum\limits_{k=m}^{n}M_k|<\varepsilon

Thus, \sum\limits_{k=1}^{\infty}f_k(x) converges uniformly.

\Box

Reflection: I’m really glad I did this proof. This has helped me remember the Weierstrass M-Test so I can use it on the qual if I need to, but it’s also a really good technique to know. When trying to show something converges uniformly you just need to show that the sequence of partial sums is Cauchy for every x in your domain.

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This entry was posted in Analysis, Cauchy, Math, Sequence, Series of Functions, Uniform Convergence. Bookmark the permalink.

One Response to Winter 1995

  1. Pingback: Everywhere Continuous Non-derivable Function | My Digital NoteBook

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