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A Suggested Rearrangement of the Book-Work on Some Elementary Series
Published online by Cambridge University Press: 03 November 2016
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In all the English text-books on Analytical Trigonometry, so far as I know, the power-series for cos x; is obtained as the limit, when n tends towards infinity, of the finite sum for cos x,
where 
and the number of terms is equal to the integer next greater than 
. The power-series for sin x is obtained similarly; the two series and the method of finding them being both due to Euler.
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- Copyright © Mathematical Association 1904
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page 85 note * It is perhaps worth while to enter a plea for the use of the phrase, from the very beginning, in dealing with limits. The phrases “the limit when n is ∞,” “the limit when n = ∞” can do nothing bnt harm.
page 85 note † Castle’s Manual of Practical Mathematics (Macmillan, 1903)Google Scholar; Lachlan, and Fletcher’s, Elements of Trigonometry (Arnold, 1904)Google Scholar. I refer to these as examples, merely; in many other respects both books strike me as good.
page 85 note ‡ I have found a little numerical calculation very useful in convincing the unbeliever that there is an assumption involved. Thus with n even, and 
, we find the product to be less than 1/103 when n = 10; less than 1/1010 for n=30; less than 1/1020 for n=50.
page 86 note * For the purpose of numerical calculation the second form is often more useful than the series.
page 87 note * The angle is taken (as usual in the Calculus) to lie between – and and π.
page 87 note † They also serve to show that the convergence to the limit is very slow, compared with that of the exponential series.
page 87 note ‡ Some of the results (perhaps all of them) are given as exercises in Prof. Gibson’s excellent Calculus; I had, however, obtained them independently before his book was published.
page 87 note § It is perhaps going too far at present to suggest that the theory of double limits and uniform convergence is essential for a professed mathematician in England; just as, for the present, it seems hopeless to expect even an elementary knowledge of the foundations of geometry from our “up-to-date” Euclids.
page 88 note * It may not be out of place to refer to the fact that the real difficulty in finding the infinite product is not to prove that x(1 – x 2/π 2), etc. are factors; but to prove that no factor of the form, eax is present. This remark is due to Stolz, but is not made in any of the English text-books.