Limit Comparison Test

Return to the Series, Convergence, andSeries Tests starting pageReturn to the List of Series TestsLet b[n] be a second series. Require thatalla[n] and b[n] are positive.If the limit of a[n]/b[n] is positive,then the sum of a[n] convergesif and only if the sum of b[n] converges.If the limit of a[n]/b[n] is zero, and thesum of b[n] converges, then thesum of a[n] also converges.If the limit of a[n]/b[n] is infinite, andthe sum of b[n] diverges, then thesum of a[n] also diverges.Here we are comparing how fast the terms grow. If the limit is positive,then the terms are growing at the same rate, so both series converge or divergetogether. If the limit is zero, then the bottom terms are growing more quicklythan the top terms. Thus, if the bottom series converges, the top series,which is growing more slowly, must also converge. If the limit is infinite,then the bottom series is growing more slowly, so if it diverges, the otherseries must also diverge.As an example, look at the seriesand compar...