Let a pie be divided into halves. Let one of the halves be divided in two, giving two quarters. Then let one of the two quarters be divided into two eighths. No matter how the pieces are rearranged, they should all fit back into the pie plate and exactly fill it. The danger of infinite series is not that they have an infinite number of terms but that they may sum to infinity.
Safety is assured if the sum of the absolute values of the terms is finite.
Such a series is called absolutely convergent.