Applying the M-Test

Real Analysis · Axiom Academy

Proving converges uniformly on with the Weierstrass M-Test. Show that the function series converges uniformly on all of . Left: every term is trapped under the cap , no matter what x is, because . Right: the cap series is finite — its partial sums climb to and stop. A finite ceiling that holds for all x at once is exactly what forces uniform convergence. You've applied the Weierstrass M-Test from start to finish. The moves to remember: The M-Test strategy: find constants M_n with for all x , then show converges. Trigonometric bound: (and likewise ) is the inequality that makes the term bound work. The p -series test: converges when ; here p = 2 , and in fact . Independence from x : the bound does not depend on x — that is exactly what upgrades pointwise to uniform convergence. Why it matters: uniform convergence lets you integrate and differentiate the series term by term under mild conditions. The M-Test is one of the most practical tools in real analysis: whenever a series of functions has bounds you can control independently of x , reach for it.

This is the written version of the interactive lesson above. See the full Real Analysis course.