Monotone Convergence

Monotone Convergence Theorem

The Monotone Convergence Theorem is a fundamental result: every bounded monotone sequence converges.

Monotone Sequences

A sequence ((a_n)) is:

• Monotonically increasing if (a_{n+1} \ge a_n) for all (n)
• Monotonically decreasing if (a_{n+1} \le a_n) for all (n)
• Strictly increasing/decreasing if the inequalities are strict

The Theorem

If ((a_n)) is increasing and bounded above, then (\lim_{n \to \infty} a_n = \sup{a_n}).

If ((a_n)) is decreasing and bounded below, then (\lim_{n \to \infty} a_n = \inf{a_n}).

Example: (a_n = 1 - 1/n)

This sequence is increasing (each term is larger) and bounded above by 1. By the theorem, it converges to (\sup{1 - 1/n} = 1).

Numerical Detection

We can check monotonicity by comparing consecutive terms:

def is_increasing(f, n):
    return all(f(k+1) >= f(k) for k in range(1, n))

Your Task

Implement:

1. is_monotone_increasing(f, n) -- checks if (f(k+1) \ge f(k)) for (k = 1, \ldots, n-1)
2. is_monotone_decreasing(f, n) -- checks if (f(k+1) \le f(k)) for (k = 1, \ldots, n-1)
3. estimate_limit(f, n) -- returns (f(n)) as an estimate of the limit using a large (n)