Newton's Infinite Summation: A Story of Discovery

While studying mathematics, a young Newton ingeniously solved the problem of finding the area of a circle using infinite series through analogy and observation. Instead of directly tackling the circle, he investigated the areas under a series of more general curves. By observing patterns in these areas, he deduced an infinite series representation for the area of a circle. This process was full of guesswork and verification. Newton not only found an approximation for the area but also developed the powerful theory of power series, laying the foundation for the development of calculus. This showcases Newton's extraordinary mathematical intuition and problem-solving skills, and teaches us that changing our perspective on a problem can often lead to greater breakthroughs.