Summing curious, slowly convergent, harmonic subseries

The harmonic series diverges. But if we delete from the harmonic series all terms whose denominators contain any string of digits such as "9", "42", or "314159", then the sum of the remaining terms converges. These series converge far too slowly to compute their sums directly. We describe an algorithm to compute these and related sums to high precision. For example, the sum of the series whose denominators contain no "314159" is approximately 2302582.33386. We explain why this sum is so close to 106 log 10 by developing asymptotic estimates for sums that omit strings of length n, as n approaches infinity. \ud \ud The first author is supported by a Rhodes Scholarship

Fun With Fourier Series

By using computers to do experimental manipulations on Fourier series, we construct additional series with interesting properties. For example, we construct several series whose sums remain unchanged when the nth term is multiplied by sin(n)/n. One series with this property is this classic series for pi/4: pi/4 = 1 - 1/3 + 1/5 ... = 1*(sin(1)/1) - (1/3)*(sin(3)/3) + (1/5)*(sin(5)/5).... Another example is sum (n = 1 to infinity) of (sin(n)/n) = sum (n = 1 to infinity) of (sin(n)/n)^2 = (pi - 1)/2. This material should be accessible to undergraduates. This paper also includes a Mathematica package that makes it easy to calculate and graph the Fourier series of many types of functions.Comment: This revision contains new results (for example, Sections 3 and 14), and Mathematica code to allow readers to check the result

Summing the curious series of Kempner and Irwin

In 1914, Kempner proved that the series 1/1 + 1/2 + ... + 1/8 + 1/10 + 1/11 + ... + 1/18 + 1/20 + 1/21 + ... where the denominators are the positive integers that do not contain the digit 9, converges to a sum less than 90. The actual sum is about 22.92068. In 1916, Irwin proved that the sum of 1/n where n has at most a finite number of 9's is also a convergent series. We show how to compute sums of Irwins' series to high precision. For example, the sum of the series 1/9 + 1/19 + 1/29 + 1/39 + 1/49 + ... where the denominators have exactly one 9, is about 23.04428708074784831968. Another example: the sum of 1/n where n has exactly 100 zeros is about 10 ln(10) + 1.007x10^-197 ~ 23.02585; note that the first, and largest, term in this series is the tiny 1/googol.Comment: 17 pages + 56 pages of Mathematica code for both Kempner and Irwin sum

The Crystal and Molecular Structure of Pentacene and Hexacene (Part 1) and of Some Derivatives of Tropolone (Part 2)

The crystal structures of the polycyclic aromatic hydrocarbons pentacene and hexacene have been determined by X-ray analysis. Pentacene is triclinic, space-group P1-, with two centro-symmetric molecules per unit cell. The molecular arrangement is similar to that in tetracene and to that in monoclinic anthracene. Although 19 carbon atoms out of 22 in the asymmetric unit are resolved in the ( Okl ) projection attempts to refine the structure by two-dimensional Fourier series methods have failed. Hexacene is also triclinic, space-group P1-, with two molecules per unit cell. The crystal structures of the hydrocarbons are closely similar to each other and to those of the preceeding members of the homologous series. A two-dimensional analysis of alpha-monobromotropolone has been carried out. alpha-Monobromotropolone is orthorhombic, space-group P21 21 21, with four molecules per unit cell. A Patterson analysis determined the positions of the four bromine atoms in the (hkO) projection. This was followed by two-dimensional Fourier syntheses and in the final projection all the atoms are clearly resolved. Bond lengths have not been worked out because the tropolone ring appears to deviate slightly from a planar heptagon. This deviation may be due to a steric effect of the large bromine atom or the deviation may be a spurious effect resulting from the effect of the heavy bromine atom on the measured intensities. An approach distance of length 3.05 A. occurs between oxygen atoms of two different molecules and might provide a basis for identifying these oxygens as hydroxyl oxygens. An analysis of cupric nootkatin has been accomplished, also in two dimensions only, by making use of the phase-determining power of the copper atom. Cupric nootkatin is monoclinic, space-group P21/a, with two centro-symmetric molecules of the co-ordination complex per unit cell. In the final projection, made by the usual Fourier series method, only 6 carbon atoms out of the 15 carbon atoms and 2 oxygen atoms in the asymmetric unit are resolved. It has proved possible, however, to elucidate the unknown aspects of the chemical structure of nootkatin viz. the points of attachment of the two side chains to the tropolone ring. Nootkatin would appear to be beta-isopropyl gamma(3-methyl but-2-enyl) tropolone. Preliminary investigations of the crystal structures of tropolone, tropolone hydrobromide, and ferric tropolone have been made. Tropolone is monoclinic, space-group P21/a, with four molecules per unit cell. Tropolone hydrobromide is monoclinic, space-group P2 1/a, with eight molecules per unit cell. The positions of the bromine atoms in the (hkO) projection have been determined, somewhat approximately because of overlap of the two bromine atoms in the asymmetric unit, by Patterson analysis. Ferric tropolone is monoclinic, space-group Cc, with four molecules of the co-ordination complex per unit cell

Long Memory and FIGARCH Models for Daily and High Frequency Commodity Prices

Daily futures returns on six important commodities are found to be well described as FIGARCH fractionally integrated volatility processes, with small departures from the martingale in mean property. The paper also analyzes several years of high frequency intra day commodity futures returns and finds very similar long memory in volatility features at this higher frequency level. Semi parametric Local Whittle estimation of the long memory parameter supports the conclusions. Estimating the long memory parameter across many different data sampling frequencies provides consistent estimates of the long memory parameter, suggesting that the series are self-similar. The results have important implications for future empirical work using commodity price and returns data.Commodity returns, Futures markets, Long memory, FIGARCH

Strengthening the Baillie-PSW primality test

The Baillie-PSW primality test combines Fermat and Lucas probable prime tests. It reports that a number is either composite or probably prime. No odd composite integer has been reported to pass this combination of primality tests if the parameters are chosen in an appropriate way. Here, we describe a significant strengthening of this test that comes at almost no additional computational cost. This is achieved by including in the test what we call Lucas-V pseudoprimes, of which there are only five less than $10^{15}$.Comment: 25 page