A series of functions is the sequence of its partial sums, so everything proved about sequences of functions applies to it at once. What is new is a way of establishing uniformity without ever computing a worst error: if each term is bounded by a number, and those numbers form a convergent series, then the series of functions converges uniformly. It is the comparison test of chapter five, made to deliver uniformity as well as convergence.

A stack of boards, each thinner than a listed thickness, with the listed thicknesses adding to a finite total. Then the stack has a finite height, and the height of what remains after the first few boards is small no matter which boards they were — which is the uniformity.

The bars are the numbers $M_k$ bounding $\lvert f_k(x)\rvert$ at every point. If they add to something finite, the series of functions converges uniformly — and the tail beyond any stage is as small as the tail of the numbers.

Let $\sum f_k$ be a series of functions on $A$ and suppose $\lvert f_k(x)\rvert\le M_k$ for every , with convergent. Then converges uniformly and absolutely on . The proof is the Cauchy criterion for uniform convergence: a block of the series of functions is bounded by the matching block of the series of numbers, which is small.

• The bounds must not depend on the point; that is what makes the conclusion uniform.
• It gives uniformity without the limit being known, since the Cauchy criterion names no limit.
• The test is sufficient and not necessary: a series can converge uniformly without the sizes of its terms being summable.
• Combined with the last three lessons: a uniformly convergent series of continuous functions has a continuous sum, and may be integrated term by term.
• $\sum\tfrac{\sin(kx)}{k^2}$ converges uniformly on , by , so its sum is continuous everywhere.

Use the Cauchy criterion, so that no limit has to be produced. A block of consecutive partial sums of the series of functions is a sum of finitely many terms, and its size at any point is at most the sum of the matching bounds, by the triangle inequality. That last quantity is a block of the convergent series of numbers, so it is small past some stage — and the stage does not depend on the point, since the bounds do not. That is exactly the Cauchy criterion for uniform convergence.

The triangle inequality on a finite sum.

Each term is bounded by its number, at every point.

The Cauchy criterion for the series of numbers, from chapter five.

The bounds do not depend on the point, so neither does the stage.

Which is the Cauchy criterion for uniform convergence.

The bound must work at every point, and the numbers must add to something finite.

Uniformity is the hypothesis every theorem of this chapter wants.

• A uniform limit of continuous functions is continuous.
• A uniformly convergent series may be integrated term by term.
• Differentiation still needs the hypothesis on the derivatives.

• The M-test bounds each term by a number valid at every point.
• If those numbers form a convergent series, the series of functions converges uniformly and absolutely.
• It needs no knowledge of the sum, being an application of the Cauchy criterion.
• It is sufficient and not necessary.

Great work! You now know how to:

• State the M-test and say why its bounds must be numbers
• Apply it to a trigonometric series
• Say what it gives besides uniformity, and what it does not
• Combine it with the theorems of this chapter