CHAPTER VI

EXPONENTIAL SERIES, LOGARITHMIC SERIES, AND
DE MOIVRE'S THEOREM

694. Expand a in a series of ascending powers of a Write a [1+ (a1)]"; expanding by the binomial theorem, we have

(1) ...

= 1 + x [ ( a − 1 ) — ! ( a − 1 ) 2 + '}'(a—1)3 — ¦ (a−1)' + · · ]

terms involving 2, 3, and higher powers of x.

Series (1) is convergent provided

-1<a-1 <1.

[681]

Equation (1) shows that a can be expanded in a series beginning with 1 and proceeding in ascending powers of x; assume, therefore, that

where 4, 4, 4,

2

are quantities not involving and therefore do not change however a may be changed; it follows from (1) and (2) that

(3)

▲, = (a — — ' −
1) — ¦ (a − 1)2 + ¦ (a − 1 ) 3 — ¦ (a−1)* + · · ·

The condition that (1) is convergent is also a sufficient condition for the convergence of (3), 2664. The coefficients A, Â ̧, etc., are easily determined by the property

37

where y is assumed to be independent of x and finite. Changing into x + y in (2), we have

(4)

a2+ y = a! +x = 1 + 4(y+x) + A2 (y+x)2 + Â ̧(y+x)3 +

but

(5)

=

2

1 + A1y +4y2 + Â ̧y3 + · ·

+ (4, + 2 4 ̧y + 3 Â ̧y2 +
+terms in higher powers of x;

a*a" = a" (1 + Â ̧¤ + Â ̧‹æ2 + Â ̧Ã3 +

3

.).c

= a + Å‚ax + Â ̧ax2 + . . . .)

The series (5) is convergent in case series (2) is convergent since the multiplier a is finite; and series (4) is convergent under the same conditions since it is equal to series (5).

Since the two expansions of a2+". are identically equal, then the coefficients of like powers of x in the two series are equal, thus

= A ̧q” = 4 ̧ (1 + Â ̧y + Â ̧y2+ Ã ̧y3 + . . . .).

[ 669]

These series are convergent since they are respectively the coeffi cients of convergent series (4) and (5); hence the coefficients of like powers of are equal (8669); thus

Series (7) is convergent for all finite values of x; since we have

Since expansion (7) is true for all values of x, take

so that

this series is usually denoted by e;

hence a

A

=la therefore α = e

4

This result is called the exponential series. Put a = e in (9), and we get an important particular case of the exponential series, thus

695. Expand log, (1 + x) in a series of ascending powers of x. In 2694, (8) we had A1=log, a which, substituted in equation (3), gives

1

e

1

1

. .

log, a = (a − 1) — ¦ (a − 1 )2 + ¦
¦ ( a − 1 ) 3 — ¦ ¦ (a − 1) ' + · · · · ·

-

.....

which is convergent when 11. Hence, if x is unity or a proper fraction, (11) may be used to calculate the value of log, (1+); but in case x is not very small the terms in the series diminish so slowly that it would be necessary to retain such a large number of them in order to secure perfect accuracy in any given decimal digit as to make the formula an impractical one for calculation. If x is greater than 1 the formula can not be used because then the series would be divergent. Hence an expansion which converges more rapidly is derived.

Subtract (12) from (11) and obtain log, (1 + x) log, (1 - x) =

-

NOTE. It has been assumed, in finding the difference of the second members, that we were dealing with sums. This is not true; we have simply the limit of sums. This step requires that the following theorem be proved: If a series is absolutely convergent (p. 648, caution and theorem) the terms of the series may be arranged as one chooses without altering the value of the series.

Substitute in the preceding equation

which is convergent for all values of n greater than 0 (Ex. XCIV, 17).

THE TABLE OF NAPIERIAN LOGARITHMS

696. The log, 2 is found by putting n=1 in expansion (13); thus

15

Though the term() has no effect on the eighth decimal place,

this does not justify the stopping of the calculation at this point. It must be shown that the remainder of the series from this point on can not influence this place either. The remainder of the series is

The value of the series in the parenthesis can not be easily calculated; but this is not necessary, because this series is evidently less than the G. P.

and therefore the remainder of the series is less than

which does not affect the seventh place. Hence we obtain for log, 2 the value

2 × .346 573 5 (9?) = .693 147 1 (8?)

or to seven places,

log, 2.693 147 2.

Log, 3 is found by putting n = 2 in expansion (13); then

Proceeding in this way the logarithm of any number to the base e can be found to any desired degree of approximation.

Similarly we obtain

log, 1= .223 143 (4?)

log, 51.609 437 (8?);

and

log, (2·5) = log, 2+ log, 5 =2.302585092

thus a table of common logarithms can be made from the table of Napierian logarithms by multiplying each logarithm by the multiplier, M.434294481 . . . ; this multiplier is called the modulus of the common system.

NOTE.-On account of the great importance of the numbers e, M, and the logarithms to the base e of 2. 3, and 5, in numerical calculation, they have been calculated to more than 260 places of decimals. (Proceedings of the Royal Society of London, Vol. XXVII, page 88.)

697. The Number Called e in Mathematics is Incommensurable. Let us suppose, for a moment, that e is equal to the commensurable number where m and n are integers. Hence we will

have

m

n

Multiply both members of this assumed equality by n!. The first member will become the integer m 1 2 3.... N 1. In the

second member beginning with 1 and ending with
are transformed into integers whose sum we
and the multiplier of the bracket becomes 1.

1

the fractions n!'

shall represent by S1, Therefore we have