A Case of Quenching Enthusiasm
In 1672, a brilliant young lawyer named Gottfried Wilhelm Leibniz was on assignment in Paris. He'd always been interested in mathematics - self-taught in geometry - and became fascinated with the intellectual energy there. The Dutch scientist and mathematician Christiaan Huygens took Leibniz under his wing and challenged his math acumen with the problem of determining what the sum of this infinite series would be:
1 + 1/3 + 1/6 + 1/10 + 1/15 + 1/21 + 1/28 + 1/36 + ...
In other words, the denominator of the nth fraction is the sum of the first n whole numbers. After some experimenting, Leibniz factored out a 2:
2[1/2 + 1/6 + 1/12 + 1/20 + 1/30 + 1/42 + 1/56 + ...]
Then, each fraction could be expressed as a difference between two numbers:
2[(1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) + (1/4 - 1/5) + (1/5 - 1/6) + (1/6 - 1/7) + ...]
Therefore, all fractions subsequently cancel each other out and we are left with 2[(1)] = 2,
thus the sum of this infinite series is 2.
Successfully solving this problem played a critical role in inspiring Leibniz to continue his investigation of such sums, and eventually to his co-creation of the Calculus, one of the greatest achievements in mathematics. Isaac Newton gets primary credit because he discovered his method of finding maxima, minima, and tangents in the 1660s, though he refused to publish it. Leibniz's methods - connecting differentiation and integration - were discovered independently a decade later, and it is his notation and terminology that is primarily used today.
It's interesting to note how this problem on an infinite series served as a motivation for moving on to an incredible accomplishment. While much has been discovered and developed since that time, it is still possible for anyone to make their own first-time discoveries in mathematics, which could then motivate more learning, and lead to more (much needed) accomplishments.
Source: Dunham, William. 1994. The Mathematical Universe: An Alphabetical Journey Through the Great Proofs, Problems, and Personalities. New York: John Wiley & Sons, Inc.