CHAPTER XVII.

EXPONENTIAL AND LOGARITHMIC SERIES.

219. IN Chap. XVI. it was stated that the logarithms in common use were not found directly, but that logarithms are first found to another base, and then transformed to base 10.

In the present chapter we shall prove certain formulæ known as the Exponential and Logarithmic Series, and give a brief explanation of the way in which they are used in constructing a table of logarithms.

220. To expand a* in ascending powers of x. By the Binomial Theorem, if n>1,

hence the series (1) is the ath power of the series (2); that is,

1

1

n

+

1 +

1

2

n

f. 40-909)

and this is true however great n may be. indefinitely increased we have

3

J

;

If therefore n be

The series

+

14

Now let ea, so that cloga; by substituting for c we

Also as in the preceding investigation, it may be shewn that when n is indefinitely increased,

that is, when n is infinite, the limit of (1 + 2) = e.

=

1

n

221. In the preceding article no restriction is placed upon the value of x; also since is less than unity, the expansions we have used give results arithmetically intelligible. [Art. 183.]

n

But there is another point in the foregoing proof which deserves notice. We have assumed that when n is infinite

for all values of r.

Let us denote the value of

~ ( ) ( ) ( )

xx

n

n

by u,.

which we have denoted by e, is very important as it is the base to which logarithms are first calculated. Logarithms to this base are known as the Napierian system, so named after Napier their inventor. They are also called natural logarithms from the fact that they are the first logarithms which naturally come into consideration in algebraical 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.

Example 2. Find the coefficient of x" in the expansion of

223. To expand log. (1 + x) in ascending powers of x.

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

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

This is known as the Logarithmic Series.

Example. If x<1, expand {log. (1+x)}2 in ascending powers of x.

By equating the coefficients of y2 in the series (1) and (2), we see that the required expansion is double the coefficient of y2 in'