This convergence check offers a option to decide the convergence or divergence of an infinite sequence whose phrases are monotonically reducing and non-negative. For a sequence a the place a a 0 for all n, it states that the sequence converges if and provided that the sequence 2a converges. As a traditional instance, think about the harmonic sequence (1/n), which is thought to diverge. Utility of this check to the sequence (1/n) utilizing this check yields the sequence (2 * (1/2)) = 1, which additionally diverges, accurately predicting the divergence of the unique sequence.
Its utility lies in its capability to rework sure slowly converging or diverging sequence into sequence which are simpler to investigate. The transformation can considerably simplify the convergence evaluation in instances the place direct comparability assessments or integral assessments are troublesome to use. Historic significance resides in its early improvement and its lasting impression on the event of extra subtle convergence assessments. Moreover, it demonstrates a invaluable method in mathematical evaluation: reworking an issue into an equal, however extra manageable, type.
