Rearranging the Alternating Harmonic Series

Transcription

1 Rearragig the Alteratig Harmoic Series Da Teague C School of Sciece ad Mathematics teague@cssm.edu 00 TCM Coferece CSSM, Durham, C

2 Regroupig Ifiite Sums We kow that the Taylor series for l( x + ) is x x x x x x x l( x+ ) = x ad that this series coverges to l( ) whe x =. We will begi with the statemet that l( ) = () ow, let's rewrite the positive terms i the series for l( ) i the followig way, l( ) = + QP + QP QP QP + () 0 We have't chaged the order i which the operatios have bee doe, we have simply reamed the positive terms by rewritig each positive term i a equivalet form. Just why we did this remais to be see, but we still have a series whose sum is l(. ) l( ) ow, divide both sides of Equatio () by. The result is = QP + QP + 0QP QP + 9 Removig the paretheses, we have QP 0 +. l( ) = () Compare the terms i the right had side of Equatio () to those i the right had side of Equatio (). Are there ay terms i oe equatio that are ot also i the other? What you will otice is that the two series have exactly the same terms. The oly differece is i the order i which they are added. Yet oe adds to l( ) ad the other to l( )! If you fid this hard to believe, write a short computer program to sum the two series. the ext page, you will fid a MathCAD template that does this computatio. Both series coverge very slowly, but it should be clear that they do ot coverge to the same values. The associative property of additio is true is ot ecessarily valid for ifiite sums. Rearragig the Alteratig Harmoic Series Da Teague 00 TCM Coferece teague@cssm.edu

3 S k = ( ) k k Z k = ( ). k ( ). k. k m.. 0 m S m (. m ) (. m ) (. m) Z m M M S M (. M ) (. M ) (. M) Z M Rearragig the Alteratig Harmoic Series Da Teague 00 TCM Coferece teague@cssm.edu

4 I the regroupig above, we took oe positive term ad two egative terms ad repeated the process. The rearraged series coverges to l ( ). What other sums ca we get? terms, The two sub-series of egative terms, =, ad positive P = , are both diverget series. We ca see that diverges, sice = = which is a costat multiplied by the sum of the harmoic series which diverges. Sice diverges, so does = , which ca be compared to P = Each term of P is larger tha the correspodig term of, so the divergece of guaratees the divergece of P. Sice each sub-series diverges, we ca rearrage the series to coverge to ay umber we choose. To rearrage the series to sum to π, for example, take eough terms from P to be just larger tha π. Say Pk < π < Pk+. ow, add the first term of. P + k < + π. ow add eough additioal terms from P util the ew sum is just larger tha π. Say Pm + < π < Pm+ +. ow, add the secod term of. P + m + + < π. Sice each ew term is smaller tha the previous, the size of the jumps o either side of π get smaller ad smaller. If we cotiue this process, the resultig series coverges to π. Ufortuately, we caot write dow the series i a ice compact form like we did with l ( ). What sums ca we get by rearragig cosistet clusters of positive ad egative terms from the alteratig harmoic series. What pattered rearragemet would give us a sum of 0? What would the sum be if we took three positive terms ad two egative terms, S = + + QP + QP QP + 9 QP + or two positives ad three egatives, S = + QP + QP + + QP + 0 We ca derive the value of the sum of a rearragemet of the alteratig harmoic series if the rearragemet is cosistet, positive terms the m egative terms. To do this, we eed three pieces of iformatio. QP +? Rearragig the Alteratig Harmoic Series Da Teague 00 TCM Coferece teague@cssm.edu

5 First, we ca write the first terms of the harmoic series, H, i terms of the odd ad eve terms. That is, H = + E. Secod, we eed to recogize that E = H, sice E = ad H =. = = Third, the differece i the sum of the first terms of the harmoic series ad lbg coverges to some costat. This costat is called Euler's umber ad is ofte symbolized by γ. This is simply comparig the value of the Riema summatio to the area uder the curve. Most of the error is geerated i the first 0 terms. To see this, use MathCad to evaluate lbg = 0. for = 0, 00,, 000,ad = D k = m D m k l( ) M D M D Usig these three pieces of iformatio, we ca determie the value to which addig positive terms ad m egative terms of the alteratig harmoic coverges. We eed to cleverly add zero twice! S = limdk Ekmi. k We are groupig the positives (odds) ad egatives (eves) i k groups of ad m, respectively. Rewrite S as Sk = k + dek Eki Ekm = dk + Eki Ek Ekm = Hk Hk Hkm, Rearragig the Alteratig Harmoic Series Da Teague 00 TCM Coferece teague@cssm.edu

6 usig the first two ideas above. ow, compare each of the three harmoic series i the expressio above the value of the associated logarithm. d i d i d i b g b g b g. Sk = k + Ek Ek Ekm = k + Ek Ek Ekm = H k Hk Hkm Sk = dhk lbkgi dhk lbkgi Hkm lbkmg + l k l k l km ow, take the limit as k. As k, dhk lbkgi γ, d lb H k k gi γ, ad d lb H km km gi γ, so we have S = γ γ γ + lim lbkg lbkg lbkmg. k But this last limit simplifies to b g b g b g b g b g b g b g b g S = + k k km k km k k k QP = + F QP = + H G I lim l l l l lim l l l lim l l K J m QP QP which is just S F = + H G I lbg l K J. m This formula gives the value S = lbg whe = ad m =. otice that the value of S is zero if = ad m =. So whe ca we rearrage the terms i a ifiite series? It turs out that, if a series is oly coditioally coverget as is the series for l( ), we caot arbitrarily rearrage a ifiite umber of terms the terms without possibly alterig the value of the series. For series that are absolutely coverget, alterig the order of the terms does ot affect the sum. For series that are coditioally coverget, the terms ca be rearraged to form a series that coverges to ay chose value, sice a coditioally coverget series cosists of a diverget sub-series of positive terms ad a diverget sub-series of egative terms. Refereces: Beigel, Richard, Rearragig Terms i Alteratig Series, Mathematics Magazie: Volume, umber, Pages: -, 9. Brow, Fo,.. Cao, Joe Elich, ad David G. Wright, Rearragemets of the Alteratig Harmoic Series College Math Joural: Volume, umber, Pages: -, 9. Rearragig the Alteratig Harmoic Series Da Teague 00 TCM Coferece teague@cssm.edu

7 Rearragig the Alteratig Harmoic Series Da Teague 00 TCM Coferece