Sequences and Series - Main Theorem

The Cauchy criterion provides a powerful characterization of convergence that doesn't require knowing the limit in advance. This fundamental theorem connects series convergence to the completeness of the real numbers.

TheoremCauchy Criterion for Sequences

A sequence $\{a_n\}$ converges if and only if it is a Cauchy sequence: for every $\epsilon > 0$ , there exists $N$ such that $m, n > N \implies |a_m - a_n| < \epsilon$

In other words, the terms get arbitrarily close to each other.

Remark

The Cauchy criterion is particularly useful because it tests convergence without reference to the limit value. It relies on the completeness of the real numbers: every Cauchy sequence converges to some real number.

TheoremCauchy Criterion for Series

The series $\sum_{n=1}^\infty a_n$ converges if and only if for every $\epsilon > 0$ , there exists $N$ such that $m > n > N \implies \left|\sum_{k=n+1}^m a_k\right| < \epsilon$

Equivalently, the partial sums $\{S_n\}$ form a Cauchy sequence.

ExampleUsing Cauchy Criterion

Show that $\sum_{n=1}^\infty \frac{1}{n^2}$ converges using the Cauchy criterion.

For $m > n$ : $\left|\sum_{k=n+1}^m \frac{1}{k^2}\right| = \sum_{k=n+1}^m \frac{1}{k^2} < \sum_{k=n+1}^m \frac{1}{k(k-1)}$

Using telescoping (partial fractions $\frac{1}{k(k-1)} = \frac{1}{k-1} - \frac{1}{k}$ ): $\sum_{k=n+1}^m \frac{1}{k(k-1)} = \frac{1}{n} - \frac{1}{m} < \frac{1}{n}$

Given $\epsilon > 0$ , choose $N > \frac{1}{\epsilon}$ . Then for $m > n > N$ : $\left|\sum_{k=n+1}^m \frac{1}{k^2}\right| < \frac{1}{n} < \frac{1}{N} < \epsilon$

Thus the series satisfies the Cauchy criterion and converges.

TheoremRearrangement Theorem (Riemann Series Theorem)

If $\sum a_n$ converges absolutely, then any rearrangement of the series converges to the same sum
If $\sum a_n$ converges conditionally, then for any real number $L$ (including $\pm\infty$ ), there exists a rearrangement that converges to $L$

Remark

This remarkable theorem shows that absolutely convergent series behave like finite sums (order doesn't matter), while conditionally convergent series are unstable under rearrangement.

ExampleRearranging the Alternating Harmonic Series

The alternating harmonic series $\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots = \ln 2$ .

We can rearrange it to converge to $\frac{3\ln 2}{2}$ by taking two positive terms for every negative term: $1 + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} + \frac{1}{7} - \frac{1}{4} + \frac{1}{9} + \frac{1}{11} - \frac{1}{6} + \cdots$

This demonstrates conditional convergence allows arbitrary sums through rearrangement.

TheoremBounded Partial Sums

If $\sum a_n$ is a series of positive terms, then: $\sum a_n \text{ converges} \iff \text{the partial sums } S_n \text{ are bounded above}$

This follows from the Monotone Convergence Theorem since $\{S_n\}$ is increasing.

ExampleEstimating Series Sums

For an alternating series satisfying the Alternating Series Test, the error in using $S_n$ to approximate the sum is bounded by the next term: $\left|\sum_{k=1}^\infty (-1)^{k-1}b_k - S_n\right| \leq b_{n+1}$

For example, to approximate $\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^2}$ to within 0.01, we need $\frac{1}{(n+1)^2} < 0.01$ , so $n \geq 10$ .

TheoremAbsolute Convergence Implies Convergence

If $\sum |a_n|$ converges, then $\sum a_n$ converges.

Proof: If $\sum |a_n|$ converges, then by the Cauchy criterion, for any $\epsilon > 0$ , there exists $N$ such that for $m > n > N$ : $\sum_{k=n+1}^m |a_k| < \epsilon$

But $\left|\sum_{k=n+1}^m a_k\right| \leq \sum_{k=n+1}^m |a_k| < \epsilon$ by the triangle inequality.

Thus $\sum a_n$ satisfies the Cauchy criterion and converges. $\square$

These theoretical results provide the foundation for understanding series convergence at a deeper level, beyond specific convergence tests.