called natural logarithms from the fact that they are the first logarithms which naturally come into consideration in algebraic investigations.
When logarithms are used in theoretical work it is to be remembered that the base e is always understood, just as in arithmetical work the base 10 is invariably employed.
From the series the approximate value of e can be determined to any required degree of accuracy ; to 10 places of decimals it is found to be 2.7182818284.
Ex. 1. Find the sum of the infinite series
We have
and by putting x =- 1 in the series for e^x, we obtain
hence the sum of the series is \frac{1}{2}(e +e^{-1}).
Ex. 2. Find the coefficient of x^r in the expansion of \frac{a-bx}{e^x}.
The coefficient required =
539. To expand \log_e (1 + x) in ascending powers of x. From Art. 537,
