Infinite series are not just concepts found in dusty math textbooks, but vibrant tools that can describe the world around us. From calculating the orbit of planets to the distribution of prime numbers, infinite series have a role to play in our understanding of the universe.
An infinite series is the sum of the terms of an infinite sequence. Imagine a sequence that goes on forever: a1, a2, a3, …. The infinite series is denoted as S = a1 + a2 + a3 + …
In the world of infinite series, we encounter two important concepts: convergence and divergence. A series converges if the sum approaches a specific value as more terms are added. If not, it diverges. A beautiful example is the geometric series given by:
S = a + ar + ar2 + ar3 + …
Some infinite series have become legendary in the realm of mathematics:
Fun fact: mathematicians like Euler and Ramanujan have deeply analyzed these and brought incredible insights to different domains.