Sequences

Sequences

A sequence is an ordered list of numbers indexed by the natural numbers. In real analysis, we write a sequence as ((a_n)_{n=1}^{\infty}) or simply ((a_n)).

Defining Sequences

A sequence is defined by a rule that assigns a real number to each natural number (n):

def a(n):
    return 1 / n  # The sequence 1, 1/2, 1/3, 1/4, ...

Common Sequences

Computing Terms

We can compute and examine any finite number of terms:

terms = [1/n for n in range(1, 11)]
print(terms)

Bounded Sequences

A sequence is bounded above if there exists (M) such that (a_n \le M) for all (n). It is bounded below if there exists (m) such that (a_n \ge m) for all (n).

Your Task

Implement compute_sequence(f, n) that takes a function f defining a sequence and an integer n, and returns a list of the first n terms: [f(1), f(2), ..., f(n)].

Also implement is_bounded(f, n, lower, upper) that checks whether all of the first n terms satisfy lower <= f(k) <= upper.