## Weierstrass M-Test: Definition

The **Weierstrass M-Test** is a convergence test that attempts to prove whether an infinite series is uniformly convergent and absolutely convergent on a set interval [x_{n}, x_{m}].

Let M_{n}(x) represent a nonnegative sequence of real numbers of *n* terms such that the summation of all terms in M_{n }is less than infinity. For a sequence F_{n}(x) consisting of real-valued functions or complex-valued functions on the interval [x_{1}, x_{2}], if the following inequalities hold true:

Then the sequence F_{n}(x) converges uniformly on the interval [x_{1}, x_{2}]. With the series of F_{n}(x) converging uniformly, it too converges absolutely without a condition needing to be set. The sum s(x) :

Becomes defined for all *x* within the closed interval [x_{1}, x_{2}]. For each *n* term in M_{n}(x), the Weierstrass M-Test must be held true for the convergence properties to exist on F_{n}(x).

## Weierstrass M-Test Proof

Convergence tests such as the M-Test follows the similar purpose of using Direct Comparison or Limit Comparison Tests (Ringstrom, 2011). If a larger summation, acting as the upper bound to F_{n}(x), converges, then series like F_{n}(x) and smaller must converge due to comparison.

In similar footsteps to comparison testing, we denote the summation of a series by:

Where S_{n}(x) is the sum of n-terms for *F(x)*. Then for m < n terms, we can set an equality for the difference of two sums based on how many terms were added up:

Pay attention to where the values *m* and *n* are located. Since we set m to be always less than n, the sum S_{k}(x) is set to add terms k = 1 to k = n. The sum S_{k}(x) for when *m* is implemented will always be a smaller interval since now the sum adds terms k = 1 to k = m. More terms are added up within S_{n}(x), thus S_{n}(x) has its terms k = 1 to k = m cut off and be changed to represent terms k = m + 1 to k = n.

The absolute value bars can be absorbed around sequence *F _{k}(x)* to change the equality into an inequality of less than or equals to.

We can replace the symbol representation of the absolute of sequence *F _{k}(x)* with M

_{k}. We know that the absolute difference between two sums should be less than infinity. Letting the sum of M

_{k}be less than infinity carries on that property.

The focus then goes on proving M_{k} to converge from other available tests, like the Ratio Test or the P-Series Test.

## The Big Picture from the Proof

If m gets large enough and still be less than n, then the tested M_{k} represents the right-end portion of the original series *h(x)* for its *m + 1* to *n* terms. The larger the m-value, the more precise the evaluation becomes on the right-most terms. Imagine as though you are telescoping the last few terms for their behavior on converging to one number.

If the terms *m+1* to *n* can get proven to converge, then the remaining portion of terms k = 1 to k = m must be helping the *m + 1* to *n* terms to converge.

## Limitation of the M-Test

Notice that the definition of the M-Test only has the potential to prove a series (in question) is uniformly convergent.

This test **cannot be used to show** that a chosen series *does not* uniformly converge.

Since the sequence M_{n}(x) chosen to test for F_{n}(x) does not depend on the convergence for x values, it is possible to have chosen M_{n}(x) whose series does not converge but the individual terms *do *converge. One can still use a non-convergent sequence M_{n}(x) and conclude F_{n}(x) to being uniformly convergent.

## Practice Problem

For a given power series below, show that *F(x)* uniformly converges for the radius of convergence spanning [-1, 1].

__Step 1__: An appropriate M_{n}(x) must be determined where all its terms are larger or equal to the terms in *F(x)*.

For a function G(x) that is F(x) ≤ G(x), G(x) = k^{-2} can be used within the interval [-1, 1].

__Step 2__: Choose a series test to determine that M_{n}(x) converges and conclude the converge of *F(x)* by the Weierstrass M-Test.

The P-Series Test states that for sequences in the form 1/k^{p}, the sequence converges if p>1.

Since p = 2 > 1, then G(x) converges and by the Weierstrass M-Test, **F(x) converges uniformly and absolutely**.

## References

Ringstrom, Hans. “A BRIEF INTRODUCTION TO UNIFORM CONVERGENCE.” *Department of Mathematics*, KTH Royal Institute of Technology, 2011, www.math.kth.se/math/GRU/2010.2011/SF1629/CTFYS/uniform.pdf.