Infinite Series: How an Endless Sum Can Equal a Finite Number

Calculators do the heavy lifting, but knowing what is going on underneath is what saves you money — or stress — later. 1/2 + 1/4 + 1/8 + 1/16 + ... = 1. Add forever, land on 1. Counter-intuitive but provable.

Convergent series

Terms shrink fast enough that the partial sums approach a finite limit.

Divergent series

Sum grows without bound — 1 + 1/2 + 1/3 + 1/4 + ... is famously divergent.

The geometric series formula

Sum of a + ar + ar² + ... = a / (1−r) when |r| < 1.

Pro tips

• The harmonic series diverges very slowly — needs ~10²⁹ terms to exceed 100.
• Taylor series expand functions as infinite polynomials — basis of computer math.

Frequently asked questions

Is 0.999... = 1? Yes, exactly equal. Different decimal representation of the same number.

What's Ramanujan's 1+2+3+... = −1/12? A special analytic continuation, not literal addition. Math YouTube oversimplifies this.

Bookmark the calculator linked above — you will need it more often than you think.