Telescoping Series

The problem, from first principles

01-geometric found a closed form by multiplying a sum by r and subtracting so that almost every term cancelled. Some series cancel without any trick at all: each term contains a piece that the next term subtracts away, like the sections of a collapsing spyglass sliding into each other until only the two ends remain. Spotting that structure turns a sum of a million terms into a single subtraction. This question implements it, and applies it to a series that does not look like it cancels until it is rewritten.

From theory to code

Implement telescoping_partial_sum(f, n), the sum of the first n terms of the series with terms f(k) - f(k + 1), then telescoping_infinite_sum(f_first, f_limit), the sum of the infinite series, then sum_reciprocal_products(n), the sum of 1 / (k * (k + 1)) for k from 1 to n. The signatures and docstrings are already in the editor.

Constraints

• f is a function taking an integer and returning a float. The series is (f(1) - f(2)) + (f(2) - f(3)) + ....
• telescoping_partial_sum takes an integer n >= 0, calls f exactly twice for any n >= 1, and never loops over the terms. With n = 0 it returns 0.0 and does not call f.
• telescoping_infinite_sum receives the value f(1) and the limit of f(k) as k grows. It does not need f itself.
• sum_reciprocal_products takes an integer n >= 0 and returns a float. It must not loop over the terms and must return exactly 0.0 for n = 0.

Hints