Weierstrass M-test

The Weierstrass M-test states that a series of functions converges uniformly if each term is bounded in magnitude by a term of a convergent series of nonnegative real numbers.

Uniform convergence: A sequence of functions converges uniformly when one index threshold controls the approximation error across the entire domain. The M-test guarantees this stronger form of convergence for the series’ partial sums.
Series of functions: A series of functions is the sequence of partial sums formed by adding functions term by term. The M-test concerns uniform convergence of these partial sums.
Term-by-term differentiation: Term-by-term differentiation differentiates the terms of a function series to obtain the derivative of its sum under suitable hypotheses. Uniform convergence of derivative series, often proved by the M-test, supports this conclusion.
Dirichlet test for uniform convergence: A uniform Dirichlet test gives uniform convergence when partial sums of one factor are uniformly bounded and another factor decreases uniformly to zero. It can prove uniform convergence without an absolutely summable termwise majorant.
Weierstrass M-test proof: A proof of the M-test bounds every tail of a function series by the corresponding tail of a convergent numerical series. The tail estimate reveals why the theorem gives uniform convergence.
Supremum norm: The supremum norm measures a bounded function by the supremum of its absolute value over the domain. The M-test bounds each term uniformly using this norm.
Term-by-term integration: Term-by-term integration integrates each term of a function series to recover the integral of its sum under suitable conditions. Uniform convergence from the M-test permits exchanging the sum and integral on bounded intervals.
Weierstrass ratio test: A ratio test for function series uses bounds on successive term ratios to establish uniform convergence. It is a specialized route to the kind of uniform convergence the M-test certifies.
Cauchy criterion for uniform convergence: A sequence of functions converges uniformly exactly when its terms satisfy a uniform Cauchy condition. The M-test makes the tails uniformly small, establishing this criterion for partial sums.
Pointwise convergence: A sequence of functions converges pointwise when it converges separately at every point in its domain. Pointwise convergence alone does not provide the uniform tail control established by the M-test.

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Abel's theorem (power series)Compared with: It offers a stronger route to exchanging limits, unlike the boundary argument needed here.