Calculus
Study of continuous change through derivatives and integrals.
Infinite Series and Convergence
Adding Up Infinite Things
Sometimes we want to add up infinitely many terms—like decimals that go on forever! Infinite series let us do this in a controlled way.
Geometric Series
A geometric series has each term multiplied by a constant ratio. For example, \( 1 + \frac{1}{2} + \frac{1}{4} + \cdots \).
\[ S = \frac{a}{1 - r} \] (for \( |r| < 1 \))
Convergence
Not all infinite series add up to a finite value! If the sum settles on a number, the series "converges." Otherwise, it "diverges."
Why Series Matter
Series are used to represent functions, calculate complicated values, and model things in engineering, physics, and even finance.
Examples
The sum \( 1 + \frac{1}{2} + \frac{1}{4} + \ldots \) converges to 2.
The decimal representation \( 0.999... \) as an infinite series equals 1.
In a Nutshell
Infinite series add up endless lists of numbers, but only some actually add to a finite amount.