## Abstract

The infinite harmonic series as hitherto understood, whether with alternating or continuous signs, is not the complete series. It may be extended in both directions to infinity, and then it exists in two forms, either with alternating or with continuous signs, both of which have finite sums. These are termed $^{A}$S$_{D}$ and $^{C}$S$_{D}$ in contradistinction to the singly infinite harmonic series $^{A}$S$_{S}$- or, since the form with alternating signs alone is convergent, S$_{S}$. Examples of both the former and their sums are given in the Epistola posterior of Newton to Oldenberg (24 October 1676): $ \matrix _{4}^{A}S_{D}^{1}\quad \text{or}\quad \sum_{M=-\infty}^{M=+\infty}\frac{(-1)^{M}}{4M+1}=1+{\textstyle\frac{1}{3}}-{\textstyle\frac{1}{5}}-{\textstyle\frac{1}{7}}+{\textstyle\frac{1}{9}}+{\textstyle\frac{1}{11}}--++\ldots \infty =\frac{\pi}{2\surd 2}, \\ _{S}^{C}S_{D}^{1}\quad \text{or}\quad \sum_{M=-\infty}^{M=+\infty}\frac{+1}{8M+1}=1-{\textstyle\frac{1}{7}}+{\textstyle\frac{1}{9}}-{\textstyle\frac{1}{15}}+{\textstyle\frac{1}{17}}-{\textstyle\frac{1}{23}}-{\textstyle\frac{1}{25}}-+\ldots \infty =\frac{\pi (\sqrt{2}+1)}{8}. \endmatrix $ The Leibniz inverse tangent series (1673), which is the special case of Gregory's general form (1671), for the magnitude of the angle, $\pi $/4, the tangent of which is unity, uniquely may be written not only as $_{2}$S$_{S}^{1}$, which of this case is $_{{\textstyle\frac{1}{2}}}$[$_{2}^{A}$S$_{D}^{1}$], but also as the complete infinite harmonic series of continuous sign with a/d = $\frac{1}{4}$, $_{4}^{C}$S$_{D}^{1}$ = $\infty \ldots ++\frac{1}{-11}+\frac{1}{-7}+\frac{1}{-3}+1+\frac{1}{5}+\frac{1}{9}+\frac{1}{13}++\ldots \infty =\frac{\pi}{4}$. The general solution for the sums, in both cases based on a well-known eighteenth-century theorem of Euler, was virtually given by Glaisher (1873) and they may be written $_{d}^{A}$S$_{D}^{a}$ or $\sum_{-\infty}^{+\infty}\frac{(-1)^{M}}{a+Md}$ = $\frac{\pi}{d}$ cosec $\frac{a\pi}{d}$; $_{d}^{C}$S$_{D}^{a}$ or $\sum_{-\infty}^{+\infty}\frac{+1}{a+Md}$ = $\frac{\pi}{d}$ cot $\frac{a\pi}{d}$. The above forms multiplied by a (or what comes to the same thing setting a as unity and using the ratio d/a for the 'common difference') bear the geometrical interpretation that the sum, for the alternating sign series, is the ratio of the length of the arc to that of the chord in a circular sector or segment of included angle (2$\pi $a)/d, and, for the continuous sign series, the same ratio multiplied by cos (a$\pi $)/d. The two series are thus completely self-contained periodic circular functions without any restriction whatever on the magnitude of the angles they numerate. Glaisher's well-known series for higher powers of $\pi $ (1873) are all doubly infinite series, as $\left[\frac{\pi}{d}\text{cosec}\frac{a\pi}{d}\right]^{2}$ = $\sum_{-\infty}^{+\infty}\frac{1}{(a+Md)^{2}}$; $\left[\frac{\pi}{d}\text{cosec}\frac{a\pi}{d}\right]^{2}\left[\frac{\pi}{d}\cot \frac{a\pi}{d}\right]$ = $\sum_{-\infty}^{+\infty}\frac{1}{(a+Md)^{3}}$; the first showing in this form that for the alternating sign series the sum of the squares of the individual terms is equal to the square of their sum. From this viewpoint the ordinary singly infinite harmonic series S$_{S}$ consists of the positive terms of a cotangent series and the negative terms of the corresponding tangent series. Nevertheless, its sum can be found by modern methods using complex numbers, when a and d are integers and a/d is not greater than unity. A table of sums for all cases up to d = 16 is included. The sums are of the form of the sum or difference of $_{\frac{1}{2}}[_{d}^{A}S_{D}^{a}]$ and a composite logarithmic quantity termed $_{d}$l$^{a}$. $ \matrix _{d}S^{a}=\frac{\pi}{2d}\text{cosec}\frac{a\pi}{d}+_{d}l^{a};_{d}S^{d-a}=\frac{\pi}{2d}\text{cosec}\frac{a\pi}{d}-_{d}l^{a}; \\ (-1)^{a}{}_{d_{0}}l^{a}=\frac{2}{d}\sum_{n=1}^{n=\frac{d-1}{2}}\cos \frac{a\pi n}{d}\log _{\epsilon}\sec \frac{\pi n}{d};\quad \quad (1) \\ _{de}l^{a_{0}}=\frac{2}{d}\sum_{n=1}^{n=d/4}\cos \frac{a\pi m}{d}\log _{\epsilon}\cot \frac{\pi m}{2d}.\quad \quad \quad \quad (2) \endmatrix $ (1) applies when d is odd, (2) when it is even, m being 2n-1. If d/2 is even, the upper limit of summation in (2) is (d-2)/4, not d/4. In (1) log sec ($\pi $n/d) is 2 coth$^{-1}$ cot$^{2}$ ($\pi $n/2d). In (2) log cot ($\pi $m/2d) is tanh$^{-1}$ cos ($\pi $m/d).