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If the capital letter a, b, c, d, e, &amp;c. be put for the numeral co-efficients of the powers of x in the said series, so that a shall be = 1, b = 1/2, c = 1/3, d = 1/4, e = 1/5, and so on, we shall have b = 1/2 x a, c = 2/3 x b, d = 3/4 x c, e = 4/5 x d, &amp;c. and the seriesx +xx /2 +x 3 /3 +x 4 /4 +x 5 /5 + &amp;c will be =x + 1/2 axx + 2/3 bx 3 + 3/4 cx 4 + 4/5 dx 5 + 5/6 ex 6 + 6/7 fx 7 + 7/8 gx 8 + &amp;c.;

philosophical transactions of the royal society of london

XLI. A method of finding, by the help of Sir Isaac Newton's binomial theorem, a near value of the very slowly-converging infinite series x + xx/2 + x 3 /3 + x 4 /4 + x 5 /5 + &amp;c. when x is very nearly equal to 1