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 give below.

Let 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 data-recalc-dims=n" />, we have the relation,

mid f_{n+1}(x)+ldots+f_m(x)mid leq M_{n+1}+ldots+M_m,~for~xin D.

Since sum M_n is convergent so there exists some nprime in N such that,

mid M_{n+1}+ldots+M_m mid leq epsilon, ~ forall m  data-recalc-dims=nprime." />

The above relations imply,

mid f_{n+1}(x)+ldots+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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