Plane trigonometry, by S.L. Loney.

LOGARITHMIC SERIES.

301

257. If y = 1, the series (3) of the previous article is equal to
1 2- 3...
which is known to be convergent.
If y= -1, it equals -1- 2 3 4.. which is known to be divergent.
In addition therefore to being true for all values of y between - 1 and
+1, it is true for the value y=l; it is not however true for the value
y=-1.
258. Calculation of logarithms to base e.
In the logarithmic series, if we put y = 1, we have
2 I 4
loge2=1-2+|-4+...... (1).
If we put = 2'
we have
loge 3 - loge 2 = log, = loge(1 +
1 1 I 1 1 1 1
2 2'22 3.23 4.24
If we put y=,
we have
loge 4 -oge3loge l 1+ I3 3 + 33. 34 +.................. (3).
From these equations we could, by taking a sufficient
number of terms, calculate loge 2, log 3, and loge 4.
It would be found that a large number of terms would
have to be taken to give the values of these logarithms to
the required degree of accuracy. We shall therefore
obtain more convenient series.