Why do some infinite series add up to a finite number?

An infinite series can add to a finite number when the terms shrink fast enough. The cleanest example is a geometric series: $$\frac12+\frac14+\frac18+\frac1{16}+\cdots.$$ Imagine filling half a square, then half of what remains, then half of what remains again. You keep adding area, but the total never passes the whole square. The sum is: $$1.$$ The formula is: $$a+ar+ar^2+\cdots=\frac{a}{1-r},\qquad |r|<1.$$ Here $a=\frac12$ and $r=\frac12$, so: $$\frac{1/2}{1-1/2}=1.$$ The important warning is that terms going to zero is necessary but not enough. The harmonic series $$1+\frac12+\frac13+\frac14+\cdots$$ still diverges. The terms must shrink fast enough for the accumulated total to settle.