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Divergent Sums in Polar form of odd/even term Formal Powerseries - Ivars - 06/21/2008
I tried to plot finite powerseries parts of several infinite divergent series by plotting separately sums with odd and even number of terms. It t is polar angle, to plot series 1-1+1-1.. I plotted r(t)=1-t+t^2-t^3+t^4 ..........+t^(2n)and r(t)=1-t+t^2-t^3+t^4-t^5 ......- t^(2n+1) the resulting series were symmetric and their combined graphs showed obvious inclination to converge towards 1/2 on vertical axis which is a "value" usually asigned to these osciullating series. Given the limited power of mu system, I only usead up to 100 terms in each series, so pictures are still quite far from convergence point. In similar way, I tried to plot: 1+1+1+... =-1/2 1-2+4-8+ =1/3 1-2+3-4+5.... = 1/4 1+2+3+4+5.. = -1/12 They all show clear tendency to converge graphically to these values polar plots. [attachment=381] The last series (-1/12) seems to need much more terms than I could produce to be sure where they converge. In this plot, all mentioned series are plotted together, but no decription on graph- this is very easy to duplicate. After that I tried plotting harmonic series. Alternating ones converged to ln2, as they should, divergent harmonic show after 100 terms makes me think that they either have value y= -ln(2) but these need much more terms to be evaluated, if possible. It might be that thses series have both r and phase value in sum. [attachment=382] I have rotated argument by replacing t with t-pi/2+1 to see if it helps to see convergence on y axis.In last 2 graphs I have added series in 1/t as well- that seems to improve localization of sum on graph as graphs now converge to that point from 4 directions. Invserse prime summation seems to lead to sqrt(3) -1 as mod ® so summation can be (-(sqrt(3)-1)) plus phase, but again with my limited precision not easy to be sure, as well as i havent found out how to really calculate exact phase- though numerically it must be possible to evaluate very accurately. [attachment=383] This polar presentation seems a good way to show graphically both convergent and divergent infinite sums. The phase seems to be an important part and feature of especially divergent summation. Ivars |