Convergence Tests for Series

A Toolkit for Real and Fourier Analysis

Determining the convergence of an infinite sum is the most frequent task in analysis. Below are the most critical tests used to establish the validity of Fourier representations and operator limits.

Direct Comparison

|a_n| \leq b_n

If \(\sum b_n\) converges, then \(\sum a_n\) converges absolutely.

Limit Comparison

\lim_{n \to \infty} \frac{|a_n|}{b_n} = c

If \(0 < c < \infty\), both series converge or diverge together.

Ratio Test

\limsup_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = L

Absolute convergence if \(L < 1\); failure if \(L > 1\).

Root Test

\limsup_{n \to \infty} \sqrt[n]{|a_n|} = L

Often superior to the Ratio Test for power series. Convergence if \(L < 1\).

Advanced Criteria

Cauchy Criterion

A series \(\sum a_n\) converges if and only if for every \(\epsilon > 0\), there exists \(N\) such that for all \(n > m \geq N\):

\left|\sum_{k = m}^{n} a_k\right| < \epsilon

This implies that the "tail" of the series must vanish at the limit.

Weierstrass M-Test

Crucial for functional sequences. If \(|f_n(x)| \leq M_n\) and \(\sum M_n\) converges, then:

\sum f_n(x) \text{ converges uniformly and absolutely.}

Dirichlet's Test

Useful for Fourier series. \(\sum a_n b_n\) converges if \(\sum a_n\) has bounded partial sums and \(b_n \to 0\) monotonically.

While these tests provide the "how" of convergence, Fourier Analysis often requires us to look at where a series converges (pointwise vs. uniform), which is where the Weierstrass M-Test becomes the primary tool.

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