Summing The Curious Series of Kempner and Irwin

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1 . INTRODUCTION. Summig The Curious Series of Kemper a Irwi Robert Baillie I 94, Kemper prove [3] that if we elete from the harmoic series all terms whose eomiators have the igit 9 that is, /9, /9, /9,..., the the remaiig series coverges. This is surprisig because it appears that this process removes oly every 0 th term from the harmoic series, so the sum of what remais woul seem to iverge. Noetheless, Kemper's series coverges to about []. The covergece is so slow that the sum of all terms with eomiators < 0 30 is still less tha. The followig geeralizatio also hols: if we elete from the harmoic series all terms whose eomiators have ay set of umbers with oe or more igits, we also get coverget series. Schmelzer a Baillie [5] show how to compute sums of these slowly-covergig series. I 96, Irwi [] geeralize Kemper's result i a ifferet way. He showe that we also get coverget series if we sum over those that have at most a fie umber of occurreces of oe or more igits. It seems that, util ow, o oe has compute sums of these series. This ote shows how to o so. For eample, we ca calculate that the sum of / where has eactly oe 9 is about It is surprisig that this is larger tha the sum with o 9, give above, because the series with oe 9 begis /9 + /9 +..., while the series with o 9 begis / + / /8 + / The partial sums of the series with oe 9 remai less tha the full sum with o 9 util we reach terms whose eomiators have 70 igits. There are i i N i 9 itegers with i igits that have eactly oe 9, so the 'oe 9' series ees i more tha 66 N N terms to ecee the sum of the 'o 9' series. A more complicate eample: the sum of / where has at most oe, two 's, three 3's, four 4's a five 5's with o coitios place o the other igits is about The sum where has eactly oe, two 's, three 3's, four 4's, a five 5's is about All sums are roue to the umber of places show.. THE ALGORITHM. It's easy to a terms havig eomiators up to, say, 7 igits, but we must use etrapolatio to get much beyo that poit. It turs out that we ca use sums over eomiators with i igits to compute the eee sums over eomiators with i + igits. The we repeat the process. The basic iea goes back to Kemper, a it is the key to accurately computig these sums. We will illustrate the metho whe there are two coitios for / to be iclue i the series: we show how to compute the sum of / where has eactly occurreces of the igit a occurreces of. The iea etes i the obvious way to coitios o three or more igits, a easily geeralizes to bases other tha 0.

2 Defie S i, to be the set of positive itegers with i igits that have occurreces of a k occurreces of. We ca geerate the set S i+, from S i,, S i, k,, a S i,, k, as follows: a For each i S i, multiply by 0, the a. b For each i S i, k, k, multiply by 0, the a. c For each i S i,, k, multiply by 0, the a 0,,,..., 9, ecept for a. Step a starts with a i-igit umber havig - occurreces of a appes as the fial igit, formig a i+ igit umber havig occurreces of. Step b oes the same for k a. I step c, the i-igit umbers alreay have the esire umber of a, so i this step, we create i+-igit umbers by appeig all other igits ecept a. Together, these steps geerate S i+,, k. If is 0, we omit step a. If k is 0, we omit step b. Net, efie t i, k, k. We will show how to compute t i+, k, k by usig i, k, k the values of t i,, k, t i, k,, a t i,, k for,, 3,.... If is a i-igit umber a is ay igit, the the reciprocal of the correspoig i+-igit umber 0 + ca be epae i powers of / : This epasio is vali because 0 <. A similar epasio hols for higher powers: 0 0! + 0!! !!! 0 Now, recallig step a, we sum these epasios for all i S i,, k. Call this sum A. A +! S i k k S i k k + 0 +!! 0,,,, 0

3 0 +!!! 0 i, k, k Likewise, summig over the sets escribe i steps b a c, we get +! i, k, 0 + 0!! 0 k i, k, k B +! i, k, 0 + 0!! 0 k i, k, k C 3 The sum i 3 is over igits,. Together, the sets geerate by steps a - c comprise S i+,. Also, A + B+ C t i+,. We have ust compute a eee sum over i+ igit umbers by usig sums over i-igit umbers. Programmig etail: we take 0 0. I orer to compute t i+,, we use the values of t i, a t i, k, k. But t i,, i tur, was compute usig t i, k a t i, k, k. This meas that, for each i a i orer to compute t i+,,, we must compute all + + values of t i, for 0 k a 0 k. If we a the t i,,, values over all i, the we get the sum of / where has eactly occurreces of a eactly occurreces of. If we a the t i,, values over all i, a over all a k with 0 k a 0 k, the we get the sum of / where has at most occurreces of a at most occurreces of. If there are three coitios,,,, 3, 3, the proceure is similar, ecept that we have three steps i place of a a b above, a that for each i a we must compute values for the t array. The time a memory requiremets icrease accorigly. Still, o a persoal computer, we ca specify a coitio for each of the te igits, provie the prouct is ot too large. 0 + To summarize, our calculatio goes as follows. We start by calculatig t i, for i,, a 3, by eplicitly aig the terms whose eomiators are i the sets S i,, for all 0 k a 0 k. For i a, we ee oly. For i 3, we compute all sums t i, for J, where J epes o the umber of ecimal places esire. The we use equatios - 3, alog with the t 3, values to successively compute the eee t i+, for i 3, 4, 5,.... We cotiue util the t i, values become small eough to be eglecte. Of course, the more ecimal places we wat, the larger the rage of i a values we will ee. 3

4 3. CONFIRMING THE CALCULATIONS. Geerally, the series cosiere here coverge very slowly. However, there are a few special cases where they coverge rapily. These special cases ca serve as a check o the algorithm. For eample, the sum of / where has o 0 i base is / This series coverges rapily to Likewise, the sum of / where has a sigle i base, is equal to. Oe ca also costruct special cases i other bases. We ca also test the etrapolatio proceure as follows. First, for a give set of coitios,,,,..., we eplicitly compute the sums over eomiators of, say,,,..., 7 igits. The, we eplicitly compute the sums through, say, 4 igits, the use equatios - 3 to estimate the sums for 5, 6, a 7 igits. I all cases, we get goo agreemet. This algorithm prouces goo agreemet with Weisstei's result i Eample b below. Fially, whe the coitios specify oly the absece of oe or more igits that is,... m 0, the results match those obtaie by the algorithms i [] a [5]. 4. MORE EXAMPLES. Eample a. Set Because we are limitig the umber of occurreces of every igit, this series has oly a fiite umber of terms. I every eomiator of this series, each igit occurs eactly oce. This meas the eomiators have eactly 0 igits, all istict. There are 0! 9 / such umbers. The sum of this fiite series is about Eample b. I orer obtai the sum i a, we ha to compute the 0 sums over those which have either 0 or occurrece of each of the te igits. Together, these comprise the positive itegers that have istict igits. There are of them betwee a The sum of their reciprocals is about This is a more precise aswer to Mothly Problem E533 tha the bous give i [4]. Iterestigly, Weisstei [7] use Mathematica to compute the eact value of this fiite sum, a fractio whose umerator a eomiator have a igits. Eample a. Set Here, each eomiator has eactly two of every igit, so all eomiators i this fiite series have 0 igits. This sum is about Eample b. As part of the calculatio i a, we also compute the 3 0 sums of / where has eactly 0,, or occurreces of each of the te igits. Together, these sums comprise a fiite sum that termiates after 0-igit eomiators. Aig these sums together gives the sum of / where has at most two of every igit. This sum is about

5 Eample 3. Wahwa [6] cosiers s k sum of / where has eactly k 0's. He shows that s k is strictly ecreasig a that s k > 9.8. We calculate that s , s , s , s ,..., s ,.... Notice also that s 0 0 l0 0. Query: why is this ifferece so small? Eample 4. Eample 3 suggests how to costruct otrivial, coverget subseries of the harmoic series that have arbitrarily large, but computable, sums. For eample, the sum of / where has at most 43 0's is about Replacig "43" i the last setece with ay larger umber yiels a coverget series with a eve bigger sum. Eample 5. Here we isplay, for each igit, the sum of / where has zero, oe, or two. Sum for Zero Occurreces Sum for Oe Occurrece Sum for Two Occurreces This table also shows, for eample, that the sum of / where has at most two 9's is about A Mathematica package implemetig this algorithm ca be owloae from [8]. Refereces. [] R. Baillie, Sums of Reciprocals of Itegers Missig a Give Digit, this MONTHLY [] F. Irwi, A Curious Coverget Series, this MONTHLY [3] A. J. Kemper, A Curious Coverget Series, this MONTHLY [4] E. S. Poiczery, Elemetary Problem E533, this MONTHLY Solutio: [5] T. Schmelzer a R. Baillie, Summig a Curious, Slowly Coverget Series, this MONTHLY 5 Jue/July

6 [6] A. D. Wahwa, Some Coverget Subseries of the Harmoic Series, this MONTHLY [7] Weisstei, Eric W. "Digit." From MathWorl--A Wolfram Web Resource. [8] The coe will be submitte to Meawhile, the coe is available from the author upo request. Remcom, Ic., 35 South Alle Street, State College, PA 680; 6

Incremental calculation of weighted mean and variance

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