Another divergence proof

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 I(a,x)=0x(sinx)acosxdx I(a,x) = \int_0^{x} \frac{(\sin x)^{a}}{\cos x} dx we will tease this until it cries out an interesting series .....

multiplying 1 continuously OR (sinx)2+(cosx)2 (\sin x)^2 + (\cos x)^2 yields =0x((sinx)acosx)((sinx)2+(cosx)2)......dx = \int_0^{x} (\frac{(\sin x)^{a}}{\cos x})((\sin x)^2 + (\cos x)^2) ...... dx =0x(sinx)acosx+((sinx)a+2cosx)((sinx)2+(cosx)2)dx = \int_0^{x} (\sin x)^{a}\cos x + (\frac{(\sin x)^{a+2}}{\cos x})((\sin x)^2 + (\cos x)^2) dx =0x((sinx)acosx+(sinx)a+2cosx+(sinx)a+4cosx.......)dx = \int_0^{x} ( (\sin x)^{a}\cos x + (\sin x)^{a+2}\cos x + (\sin x)^{a+4}\cos x ....... \infty ) dx =((sinx)a+1a+1+(sinx)a+3a+3+(sinx)a+5a+5.....) = ( \frac{(\sin x)^{a+1}}{a+1} + \frac{(\sin x)^{a+3}}{a+3} + \frac{(\sin x)^{a+5}}{a+5} ..... \infty ) plug in a=1;x=π2 a=1 ; x = \frac{\pi}{2} =1a+1+1a+3+1a+5...... = \frac{1}{a+1} + \frac{1}{a+3} + \frac{1}{a+5} ...... \infty to get =12(11+12+13......) = \frac{1}{2} ( \frac{1}{1} + \frac{1}{2} + \frac{1}{3} ...... \infty ) which is half the harmonic series ........ solving the integral for the same a=1 gives I(1,π2)=0π2tanxdx I(1,\frac{\pi}{2}) = \int_0^{\frac{\pi}{2}} \tan x dx = = \infty ....... 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!

Note by Abhinav Raichur
6 years, 1 month ago

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