Latest

A sequence asks: what's the pattern? A series asks: what does it add up to? The interplay between those two questions drives everything from Zeno's paradox to the way your calculator computes sine.

To define a Fibonacci number, you need Fibonacci numbers. That circularity isn't a flaw — it's a feature. Recursion encodes process, not just pattern, and it shows up everywhere from population models to sorting algorithms.

A polynomial has a last term. A power series doesn't — and that small change lets mathematicians represent transcendental functions with pure arithmetic. Your calculator doesn't look up sin(x); it computes a power series.

Adding infinitely many numbers shouldn't give you a finite answer — and sometimes it doesn't. Convergence tests are the tools for figuring out which situation you're in before you waste effort calculating.

1 + 1/2 + 1/4 + 1/8 + … equals exactly 2. That shouldn't work — and understanding why it does is a genuine conceptual leap in how we think about infinity.

Geometric series have a formula. And unlike arithmetic series, they can sum to finite values even when infinite. Add up 1 + 2 + 4 + 8 + 16 + 32 + 64. You could add them one by one. Or you could notice: this equals 2⁷ - 1 = 127. That's not a coincidence.

The young Gauss summed the first 100 integers by pairing opposites. That intuition scales into the general arithmetic series formula — and once you see the geometry, the algebra becomes obvious.