, ▁we ▁have ▁that ▁. ▁Since ▁ ▁is ▁bounded ▁by ▁M ▁and ▁, ▁the ▁first ▁of ▁these ▁terms ▁approaches ▁zero , ▁ ▁as ▁. ▁ ▁We ▁have , ▁for ▁each ▁k , ▁. ▁But , ▁if ▁ ▁is ▁decre asing , ▁ ▁, ▁ ▁which ▁is ▁a ▁teles cop ing ▁sum , ▁that ▁equals ▁ ▁and ▁therefore ▁approaches ▁ ▁as ▁. ▁ ▁Thus , ▁ ▁converges . ▁And , ▁if ▁ ▁is ▁increasing , ▁ ▁, ▁ ▁which ▁is ▁again ▁a ▁teles cop ing ▁sum , ▁that ▁equals ▁ ▁and ▁therefore ▁approaches ▁ ▁as ▁. ▁ ▁Thus , ▁again , ▁ ▁converges . ▁ ▁So , ▁ ▁converges ▁as ▁well ▁by ▁the ▁direct ▁comparison ▁test . ▁ ▁The ▁series ▁ ▁converges , ▁as ▁well , ▁by ▁the ▁absolute ▁convergence ▁test . ▁Hence ▁ ▁converges . ▁ ▁App lications ▁▁ ▁A ▁particular ▁case ▁of ▁Dir ich let ' s ▁test ▁is ▁the ▁more ▁commonly ▁used ▁altern ating ▁series ▁test ▁for ▁the ▁case ▁▁▁▁ ▁Another ▁cor ollary ▁is ▁that ▁ ▁converges ▁whenever ▁ ▁is ▁a ▁decre asing ▁sequence ▁that ▁tends ▁to ▁zero . ▁ ▁Im pro per ▁integr als ▁▁ ▁An ▁analog ous ▁statement ▁for ▁convergence ▁of ▁impro per ▁integr als ▁is ▁proven ▁using ▁integration ▁by ▁parts . ▁If ▁the ▁integral ▁of ▁a ▁function ▁f ▁is ▁uniformly ▁bounded ▁over ▁all ▁intervals , ▁and ▁g ▁is ▁a ▁monot on ically ▁decre asing ▁non - negative ▁function , ▁then ▁the ▁integral ▁of ▁f g ▁is ▁a ▁conver gent ▁impro per ▁integral . ▁ ▁Notes ▁ ▁References ▁▁▁