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

1+0+*+.....=(1+1+&+ +...)

2+3+1

1

The series

1+1+

t......

is usually denoted by e;

hence

ex=1+at

x2
+

+......
13.4

+

Write cx for 2, then

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Cir

ect=1+cx+

+.

12 3 Now let e=d, so that c=logea; by substituting for c we obtain

22 (logea), 2 (loge a)3 ax=1+x logea +

t....... 12

+

This is the Exponential Theorem.

+

+......

508. The series

1 I 1 1+1+

+

12' 13'14 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 1. Find the sum of the infinite series

1 1 1
+

+
12 4 16

1+

+

1 1 We have e=1+1+

+ + ......;

3.4 and by putting x= 1 in the series for ex, we obtain

1 1 1
e-1=1-1+

+
12
3

4

1 1 1 te-i=2(1+

2 46

+

+

....);

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

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-X +

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= (a – bæ) {1

(-1)"x"

2 13 T The coefficient required=(-1)" (-1)=-1

lr
(-1)"

(a +rb).

. a

[ocr errors]

lr-1

r

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

+

+

From Art. 507,

y2 (loge a)2 , y3 (loge a)3 ay=1+y loge a +

.

t....... 12

3 In this series write 1+x for a; thus (1+x)" + * {24%. logx3 ......

13

Also by the Binomial Theorem, when y<1 we have
(1+x)y=1+y +
y (y-1) y

3

23 +

...(2).

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Now in (2) the coefficient of y is
(-1) (-1)(-2)
x +

(-1)(-2)(-3)
x +

204 + ...... ; 1.2 1.2.3

1.2.3.4

x2 xt that is,

3 4

+

23

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

x2 23 xt loge (1+x)=x

2 3 4

+ This is known as the Logarithmic Series.

+

510. Except when x is very small the series for loge (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.

+

511. In Art. 509 we have proved that

X2 X3 loge (1+x)=x

2 3 changing x into – x, we have

x2 X3 loge (1 – X)=

2 3 By subtraction,

1+x
loge 2 X +

+

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1
1

1 loge (n+1) - loge n=2 +

+ (2n +1 (+13 5 1)5 From this formula by putting n=1 we can obtain loge 2. Again by putting n=2 we obtain loge 3 – loge 2; whence loge 3 is found, and therefore also loge 9 is known.

Now by putting n=9 we obtain loge 10 – log. 9; thus the value of log. To is found to be 2:30258509.... To convert Napierian logarithms into logarithms to base 10

1 we multiply by which is the modulus [Art. 413] of the

loge 10

plogo (n+1)– ulog, n=2u (2n?

;

1 common system, and its value is

or •43429448...;

2:30258509...' we shall denote this modulus by u.

By multiplying the last series throughout by u we obtain a formula adapted to the calculation of common logarithms. Thus

1

1
+
+

+.. 2n +1 that is,

р
f

M log10 (n+1) - logio n=2 +

+

+

3(2n+1)3 + 5 (2n +1)5 From this result we see that if the logarithm of one of two consecutive numbers be known, the logarithm of the other may be found, and thus a table of logarithms can be constructed.

EXAMPLES XLVIII.

+

1. Show that

23 24 25 (1) -2=1

+......

5
e2-1 1 1 1
(2)

3+
2e

t......

5. 2. Expand log V1+x in ascending powers of ..

1 1 1 1 3. Prove that loge 2 + + + st

2 12 30 42

+

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6. Show that if x>1,

1 log V3"-1=log x

2.x2 7. Show that

1 + x log

CHAPTER XLIX

DETERMINANTS.

512. CONSIDER the two homogeneous linear equations

ayx+by=0,

+buy=0; multiplying the first equation by be, the second by b, subtracting and dividing by x, we obtain aybz a,=0......

(1). This result is sometimes written

a, b, =0,

a, b,

and the expression on the left is called a determinant. It consists of two rows and two columns, and in its expanded form or development, as seen in the first member of (1), each term is the product of two quantities; it is therefore said to be of the second order. The line a,b, is called the principal diagonal, and the line b,a,, the secondary diagonal.

The letters an, bz, Az, b, are called the constituents of the determinant, and the terms a,b2, a,b, are called the elements. 513. Since

a, b, ) =ab2 a,bı= | a, a, ,
a, b2

bi b, it follows that the value of the determinant is not altered by changing the rows into columns, and the columns into rows. 514. Again, it is easily seen that

and

a, b, a, b, bg a,

a2

aq bi

=18

a b

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ba

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