Suppose we had an infinite series - z = ∑i = 1 to ∞ ( α1(i)x1 + α2(i)x2 + . . . + αm(i)xm ) - rewritten as the cumulative sequence - z(n) = α1(n)x1 + α2(n)x2 + . . . + αm(n)xm - where the xj are linearly independent and normalized (and serve as a finite basis across the sequence). If all the coefficients converged, then z(n) converges. If only one of the coefficients diverges, then z(n) diverges. How do we assess the case where more than one of the coefficients diverge (while noting that they are linearly independent)? Our space is a normed linear space.