Cauchy’s tests are used to determine the convergence of sequences and series. Here are the main ones:
- Cauchy test for sequences:
A sequence ((a_n)) converges if for any (\epsilon > 0) there exists a natural number (N) such that for all (m, n > N) the inequality holds:
[
|a_n – a_m| < \epsilon
] - Cauchy test for series:
A series (\sum_{n=1}^{\infty} a_n) converges if for any (\epsilon > 0) there exists a natural number (N) such that for all (m > n \geq N) the following holds:
[
\left| \sum_{k=n}^{m} a_k \right| < \epsilon
] - Cauchy test for integrals:
The integral (\int_a^bf(x) \, dx) converges if for any (\epsilon > 0) there exists a partition ({x_0, x_1, \ldots, x_n}) from (a) to (b) such that:
[
\sum_{i=1}^{n} |f(x_i^ )| \Delta x_i < \epsilon ] where (x_i^ ) is an arbitrary point in the interval ([x_{i-1}, x_i]).