Download Elementary Quantum Field Theory by Ernest M. Henley, Walter Thirring PDF

By Ernest M. Henley, Walter Thirring

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But it is to be noted that the final sum must be zero in both caseSj this always happens when the curve passes through the point A on the abscissa, and at the same time has ab as asymptote. This caution is to be applied so that the sums which have to be investigated by the methods presented are true, requiring no correction at all, as is very often the case in the quadrature of curves. 176) If the terms of some series are formed by writing the numbers 1, 2, 3, 4, 5, etc. ). Here it is to be noted that the quantity z + 1 is multiplied into the whole series which immediately follows it.

Corollary. If n is a negative integer or zero, the value of S will terminate, the series being summable. And where m is negative, the series will be infinitely large. But here I except the case in which m = 0, for then the series will be summable by Example 1 of Proposition 5. Example 1 Let the series t + ~A + ~B + ¥C + ~D + 151 E + &c. be proposed for summation. , and when that has been compared with the general equation it gives n = ~, m - 1 = -2, or m = -1; when these have been substituted, t, A 3B t, 5C 7D 9E S = 2T- - - - - - - - - - - - - - -&c.

Now the demonstration of this depends on the Newtonian method of differences. Let us consider the series 1, -1, 0, 8, 27, 61, 114, 190, etc. When the differences have been collected together in the manner explained above, it will be found that A = 1, B = -2, C = 3, D = 4, while the remaining coefficients are zero: -1 1 -2 1 3 ° 8 8 7 4 27 19 11 4 4 190 76 53 23 19 15 4 114 61 34 4 and so the sum comes out as z z z-I z z-I z-2 z z-I z-2 z-3 - -2 x - x - - +3 x - x - - x - - +4 x - x - - x - - x - I 12 12 3 12 3 4 40 Part I .

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