Infinite Powers: How Calculus Reveals the Secrets of the Universe

by Steven H. Strogatz

Thus the answer to Huygens’s original question about the infinite sum becomes clear: As N approaches infinity, the term 1/(N+1) approaches zero, and so S approaches 1. That limiting value of 1 is the answer to Huygens’s puzzle. The key that allowed Leibniz to find the sum was that it had a very particular structure: it could be rewritten as a sum of consecutive differences (in this case, differences of consecutive unit fractions). That difference structure caused the massive cancellations we saw above. Sums with this property are now termed telescoping sums because they call to mind one of those old collapsible telescopes you see in pirate movies, the kind of spyglass that can be stretched out or contracted at will. The analogy is that the original sum appears in its stretched-out form, but, because of its difference structure, it can be telescoped down to a much more compact result. The only terms that survive the collapse are the terms without partners to cancel them, the ones at the very ends of the telescope.