﻿ Weirstrass M-Test - Gonit Sora

## 30 Nov Weirstrass' M-Test

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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>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 >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.