Here is another proof for the divergence for the harmonic series . I assume no one has found it earlier! (its awfully dumb to say I'm the first one!!)..... consider the integral we will tease this until it cries out an interesting series .....
multiplying 1 continuously OR yields plug in to get which is half the harmonic series ........ solving the integral for the same a=1 gives ....... hence the harmonic series diverges!
What is more important here is the behavior of the series as the limit transforms from 0 to x ..... When x starts from 0 further (keeping a as 1 ) and reaches pi/2 the series changes its values according to the function tan x ....... beautiful! isn't it ?
please post your comments and ideas!