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Math Analysis 1

Sequences and their limits: definitions, theorems and methods

Definition of limit, properties, indeterminate forms, monotone sequences and the number e, recursive sequences, subsequences and the Cauchy criterion.

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A sequence is an infinite list of numbers a_1, a_2, a_3, \dots, that is a function defined on the natural numbers. It is the first place where you meet the idea of a limit, in a simpler form than for functions, and it comes back in series, numerical methods and discrete-time signals.

Definition of limit

We say that a_n converges to L, written a_n \to L, if

    \[ \forall \varepsilon > 0 \ \exists N : \ n > N \Rightarrow |a_n - L| < \varepsilon. \]

In words: whatever tolerance \varepsilon you choose, from some index on all the terms are within \varepsilon of L. The sequence diverges to +\infty if for every M, from some index on, a_n > M. If it neither converges nor diverges it is called oscillating (or indeterminate), like (-1)^n.

Example. a_n = \frac{1}{n} \to 0: given \varepsilon, take N > \frac{1}{\varepsilon}. With \varepsilon = 0.01 every term from n = 101 on will do.

Basic properties

  • Uniqueness: if the limit exists it is unique.
  • Boundedness: a convergent sequence is bounded (the converse is false: (-1)^n).
  • Sign preservation: if a_n \to L > 0, from some point on a_n > 0.
  • Algebra of limits: the limit of a sum, product or quotient (with a denominator not tending to 0) is the sum, product or quotient of the limits.
  • Squeeze theorem: if a_n \le b_n \le c_n and a_n, c_n \to L, then b_n \to L too.

Indeterminate forms

The algebra of limits says nothing in the cases \infty - \infty, 0\cdot\infty, \frac{0}{0}, \frac{\infty}{\infty}, 1^\infty, 0^0, \infty^0. There you have to work on the expression. Two frequent techniques:

  • Factor out the leading term: \dfrac{3n^2 - n}{n^2 + 5} = \dfrac{n^2(3 - 1/n)}{n^2(1 + 5/n^2)} \to 3.
  • Rationalise: \sqrt{n+1} - \sqrt n = \dfrac{1}{\sqrt{n+1} + \sqrt n} \to 0.

Monotone sequences and the number e

A central result, following from the completeness of the reals: every monotone sequence has a limit; if it is also bounded, the limit is finite (it is the supremum or infimum of its values). This is how one proves that

    \[ \left(1 + \frac1n\right)^n \to e \approx 2.71828, \]

because this sequence is increasing and bounded (by 3, for instance). It is also the model of compound interest: a 100% interest rate split into n ever more frequent instalments makes the capital grow at most by a factor e.

Recursive sequences

Often each term is computed from the previous one: a_{n+1} = g(a_n). If the sequence converges to L and g is continuous, the limit must satisfy L = g(L). Example: a_1 = 1, a_{n+1} = \frac12\left(a_n + \frac{2}{a_n}\right). The limit, if it exists, satisfies L = \frac12(L + 2/L), that is L^2 = 2: the sequence approximates \sqrt2 (1; 1.5; 1.4167; 1.41422…). It is the Babylonian method, a special case of Newton’s method that computers still use for square roots.

Subsequences and the Cauchy criterion

If a_n \to L, every subsequence tends to L. That is why finding two subsequences with different limits proves the limit does not exist. The Bolzano-Weierstrass theorem guarantees that every bounded sequence has a convergent subsequence (see the in-depth article). Finally the Cauchy criterion: a sequence converges if and only if its terms become arbitrarily close to each other, |a_n - a_m| < \varepsilon for large n, m. It is useful because it proves convergence without knowing the limit.

Limits worth remembering

  • q^n \to 0 if |q| < 1, \to +\infty if q > 1, oscillating if q \le -1;
  • \sqrt[n]{a} \to 1 for a > 0, \sqrt[n]{n} \to 1;
  • \dfrac{n^k}{a^n} \to 0 for a > 1, \dfrac{a^n}{n!} \to 0, \dfrac{n!}{n^n} \to 0.

In discrete-time signals a sequence is a sampled signal: x[n] = q^n with |q| < 1 is an echo that dies out, and its convergence to zero is the stability of the system that produces it.

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