diverging harmonic series

[AntonioLao]

Sensibly, every mathematical function can be represented as an infinite series. However, the limit of a particular partial sum S(n) may fail to exist either when S(n) increase indefinitely with n or when the partial sum S(n) oscillates without approaching a limit as n approaches infinity. For example, the series 1+1+1+1+1+…+ad infinitum diverges because its nth partial sum S(n)=n increases with n without limit. On the other hand, the series 1-1+1-1+1-1+1-1+…-1 ad infinitum also diverges because its partial sums S(1)=1, S(2)=0, S(3)=1, … oscillate between the values 1 and 0.

Interestingly, the harmonic series 1+1/2+1/3+1/4+1/5+1/6+1/7+…+1/n can be shown to diverge as n approaches infinity. If the terms are grouped as follows: (1+1/2)+(1/3+1/4)+(1/5+1/6+1/7+1/+(1/9+1/10+1/11+1/12+1/13+1/14+1/15+1/16)+…where the smallest term of each group has denominator as the increasing powers of 2 then each group is larger than a group where all larger terms are replaced by the smallest term of the group. That is to say, (1+1/2) is replaced by (1/2+1/2), (1/3+1/4) is replaced by (1/4+1/4)=1/2, and (1/5+1/6+1/7+1/ is replaced by (1/8+1/8+1/8+1/=1/2, etc. The new series is reduced to the series of ½+ ½ + ½ + ½ + ½ + … or ½ (1+1+1+1+1+1+…) which is divergent. Since the harmonic series is greater than this new series, it is also divergent. This illustrates the important idea of comparison, which is a fundamental mathematical tool for proving the convergence of infinite series.