Page:Elementary algebra (1896).djvu/451

433 Rh In this series write 1 + x for a; thus (1 + x)^r

Also by the Binomial Theorem, when y < 1 we have

Now in (2) the coefficient of y is

that is.

Equate this to the coefficient of y in (1); thus we have

This is known as the Logarithmic Series.

540. Except when x is very small the series for \log_e (1 +x) is of little use for numerical calculations. We can, however, deduce from it other series by the aid of which Tables of Logarithms may be constructed.

541. In Art. 539 we have proved that

changing x into -x, we have

By subtraction,

Put \frac{1+x}{1-x}=\frac{n+1}{n} so that x = \frac{1}{2n+1} we thus obtain