Weirstrass' M-Test

We shall state and prove a very important result in Real Analysis called the Weirstrass’ M-Test. The statement of the theorem is given below.

Let \{M_n\}_{n\geq 0} be a sequence of positive real numbers such that \mid f_n(x) \mid \leq M_n for x \in D, n \in N. If the series \sum M_n is convergent, then \sum f_n is uniformly convergent on D.

We prove the result as follows:

If m>n, we have the relation,

\mid f_{n+1}(x)+\cdots+f_m(x)\mid \leq M_{n+1}+\cdots+M_m,~for~x\in D.

Since \sum M_n is convergent so there exists some n^{\prime} \in N such that,

\mid M_{n+1}+\cdots+M_m \mid \leq \epsilon, ~ \forall m >n^{\prime}.

The above relations imply,

\mid f_{n+1}(x)+\cdots+f_m(x)\mid < \epsilon.

It is thus clear now that \sum f_n is uniformly convergent on D.

There are other variants of this result, which is left for a later post.


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