A curious result related to Kempner s series

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1 Amer. Math. Mothly, 5 (28), p A curious result related to Kemper s series Bakir FARHI bakir.farhi@gmail.com Abstract It is well kow sice A. J. Kemper s work that the series of the reciprocals of the positive itegers whose the decimal represetatio does ot cotai ay digit 9, is coverget. This result was exteded by F. Irwi ad others to deal with the series of the reciprocals of the positive itegers whose the decimal represetatio cotais oly a limited quatity of each digit of a give oempty set of digits. Actually, such series are kow to be all coverget. Here, lettig (r N) deote the series of the reciprocal of the positive itegers whose the decimal represetatio cotais the digit 9 exactly r times, the impressive obtaied result is that teds to log as r teds to ifiity! MSC: 4A5. Keywords: Kemper s series; Harmoic series; Series with deleted terms. Itroductio Throughout this article, we let N deote the set N \ {} of positive itegers. We let E (r) (r N) deote the set of all positive itegers whose decimal represetatio cotais the digit 9 exactly r times. We also let E (r) (r, N) deote the set E (r) [, + [. We clearly have for ay r N: E (r) N E (r). For all r, N, write: ad for all r N, write: : k E (r) k : k S (r). k E (r) N

2 I 94, A. J. Kemper [3] showed that the series S () coverges. After him, several geeralizatios were obtaied by several authors. Amog others, F. Irwi [2] showed that the series derived from the harmoic series by icludig oly those terms whose deomiators cotai a limited quatity of each digit of a give oempty set of digits is coverget. It follows i particular that the series coverges for all atural umbers r. I this paper, the mai result obtaied is that the sequece ( ) r decreases ad coverges to log (see Theorem 3). As a cosequece, we deduce that we have > log ( r ). So, accordig to the calculatios of R. Baillie [], we have the uexpected iequality S () < S (). We must otice that the approximate umerical values of the s (r ) are very difficult to calculate. I the last sectio of the paper, we state a geeralizatio of our mai result by takig istead of the digit 9 ay other digit d {,,..., 9}. 2 The Results Suppose r, N. If a positive iteger k belogs to E (r), the writig k t + l with t N ad l {,,..., 9}, we clearly have either t E (r) ad l {,,..., 8}, or t E (r ) ad l 9. It follows that 8 l t + l + t E (r ) t + 9. () We will fid it useful to approximate with the simpler formula T (r) : 8 l t + t E (r ) The error i this approximatio is give by t 9 S(r) + S(r ). (2) So, we have C,r : T (r). (3) 9 S(r) + S(r ) C,r. (4) 2

3 This idetity will play a importat role i what follows. Our first propositio shows that the errors C,r are ot very large. Propositio The real umbers C,r (r, N ) are all oegative ad we have C,r <. r Proof. From () ad the first equality of (2), we have for all r, N : C,r 8 l l t(t + l) + t E (r ) 9 t(t + 9). (5) This last idetity shows that we have C,r ( r, N ). Moreover, usig (5), we have for all r, N : C,r 9 25 t t E (r ) t 2. Sice the sets E (r) (r, N) form a partitio of N, it follows that: r C,r ( ) t t 9 π <. The proof is complete. For the followig, put for all r N : C r : C,r. Accordig to (5), we clearly have C,r > for all, r N such that r. Cosequetly, we have C r > for all r N. Propositio 2 For all r N, the series coverges. I additio, we have: ad S () S () C + 9 (6) S (r ) C r ( r 2). (7) 3

4 Proof. The fact that the series (r N) are all coverget is already kow (see, e.g., [2]). Let us prove the relatios (6) ad (7) of the propositio. Usig the relatio (4), we have for all r N : Hece: + ( 9 S(r) + ) S(r ) C,r S (r ) C,r + 9 S (r) + S (r ) C,r + 9 S(r) + S(r ) C r +. (r ) C r + S { S (r ) C r + 9 if r, S (r ) C r if r 2. This cofirms the required formulas (6) ad (7) of the propositio ad fiishes this proof. We ow arrive at the most importat ad completely ew result of this paper: Theorem 3 The sequece ( ) r decreases ad coverges to log. I particular, we have: > log ( r ). Proof. Sice C r > ( r ), the formula (7) of Propositio 2 shows that the sequece ( ) r decreases. Sice this sequece is positive (so bouded from below by ), it is ecessarily coverget. It remais to calculate its limit as r 4

5 teds to ifiity. We have for all iteger R 2: S (R) Hece: R ( S (r )) + S () r2 R ( C r ) + S () C + 9 r2 S () R r C r + 9. lim R S(R) S () r (accordig to (6) ad (7)) C r + 9. (8) Now, let us calculate the sum r C r r C,r. From (), (2), ad (3), we have for all r, N : C,r : T (r) 8 r l ( t t + l l t N \E () ) + t E (r ) ( ) t. t + 9 By remarkig that the sets E (r) (r, N ) form a partitio of N \ E () ad that the sets E (r ) (r, N ) form a partitio of N, we deduce that: 8 ( C,r t ) ( ) + t + l t t l 9 l 9 l t t t ( t ) ( t + l t ) t + l + ( ) t t + 9 t E () t ) 8 ( t t + l ( t ) 9 t + l Sice t E () /t S () ad l t E () t E () t + t ( t ) t + l 8 t + l. t E () l 8 l t E () t + l t E () t 8 t t S() 8 t t 5

6 (because {t + l t E (), l {,,..., 8}} E () \ {, 2,..., 8}), it follows that: C,r + 8 S() t, (9) where : 9 l t r ( t ) t + l t t t t m<(t+) m. Let us calculate. For all sufficietly large positive itegers N, we have: N t N m N+9 t m t m t t m<(t+) (log N + γ) 9 m { m log + o N(), log(n + 9) + γ 9 m } + o N () m where γ deotes the Euler s costat. By takig the limits as N teds to ifiity, we obtai: 9 log. m m Now, by substitutig this value of ito (9), we obtai: C r r r C,r S() log + 9. Fially, by substitutig this value of r C r ito (8), we coclude that: lim R S(R) log as required. The proof is complete. 6

7 3 Geeralizatio to other digits The method preseted above ca be applied to obtai the geeral result related to ay digit d {,,..., 9}. For d {,,..., 9} ad for r N, let σ d (r) deote the sum of the reciprocals of all positive itegers whose decimal represetatio cotais the digit d exactly r times. The, we have the followig: Theorem 4 For all d {,,..., 9}, the sequece (σ d (r)) r decreases ad coverges to log. Remark. Accordig to Theorem 4 ad to the approximate values of the umbers σ d () ( d 9) give by R. Baillie [], we see that oly i the particular case d (already studied by A. D. Wadhwa [4]) does the sequece (σ d (r)) r start its decrease at r. Refereces [] R. Baillie, Sums of reciprocals of itegers missig a give digit, Amer. Math. Mothly, 86 (979), p [2] F. Irwi, A curious coverget series, Amer. Math. Mothly, 23 (96), p [3] A. J. Kemper, A curious coverget series, Amer. Math. Mothly, 2 (94), p [4] A. D. Wadhwa, Some coverget subseries of the harmoic series, Amer. Math. Mothly, 85 (978), p

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