Geometric Series

The problem, from first principles

A geometric series adds up a list of numbers where each one is the previous one multiplied by the same fixed ratio: a bank balance that grows by 5% every year, a signal that loses half its strength at every step, a reward that is worth a little less with each step of delay. Adding the terms one by one works for a few terms, but it gives no way to answer the questions that matter: what is the total after a million terms, and does the total settle on a finite number if the list never ends? This question builds the closed forms that answer both.

From theory to code

Implement geometric_partial_sum(a, r, n), the sum of the first n terms, then geometric_series_sum(a, r), the sum of the infinite series when it has one, then terms_needed(a, r, tolerance), how many terms are needed before the partial sum is within tolerance of the infinite sum. The signatures and docstrings are already in the editor.

Constraints

• a is the first term and r is the common ratio, both floats. The series is a + a*r + a*r**2 + ....
• geometric_partial_sum takes an integer n >= 0. With n = 0 there are no terms, so the sum is 0.0. It must handle r == 1 without dividing by zero, and it must not loop over the terms.
• geometric_series_sum only exists when |r| < 1. For any other ratio it raises ValueError.
• terms_needed requires |r| < 1 and tolerance > 0, and returns the smallest integer n >= 0 such that the gap between the infinite sum and the sum of the first n terms is below tolerance. If a == 0 the answer is 0.

Hints