"x-x=0". Here we have two terms (x and -x). Their magnitude grows at the same rate, so it is not surprising that their difference is fixed and the two-term "series" "converges" to zero.
"Any converging series." Consider the exponential growth function: e^x = 1 + x + x^2/2 + x^3/6 ... + x^n/n!. This grows fast, but we know that the factorial is super-exponential so the terms must approach zero for any fixed x. If the series converges (true though not obvious) then the value must increase very quickly.
Now consider the exponential decay function. Unlike x-x, the terms are raised to different powers, so they grow at different rates. And unlike the exponential growth function, the series converges to zero in the limit.
The "miracle" is not the convergence but how it happens: terms growing in opposite directions at wildly different rates. It's as if a lopsided spaceship chaotically fired rockets at full power in all directions, and happened to stay exactly still.