552. We have found in Art. 550, that when n increases

nx

without limit (1+1) ultimately becomes e"; in the same way

we may shew that when n increases without limit (1 + 2 )

ultimately becomes e**.

EXAMPLES OF LOGARITHMIC SERIES.

1. Prove that log, (x+1)= 2 log, x-log, (x-1)

series to calculate log10 11.

2. Shew that log, (x + 2h) = 2 log, (x + h) — log, x

+

3. If a, b, c be three consecutive numbers,

.......

4. If A and μ be the roots of ax2 + bx + c = 0, shew that

5. Log,{1+1+x+ (1 + x)3} = 3 log. (1 + x) — log, x

9. Assuming the series for log. (1 + x) and e", shew that

nearly when n is large; and find the next term of the series of which the expression on the second side is the commencement.

10. Find the coefficient of x" in the development of

in which the successive terms are formed by some regular law, and the number of the terms is unlimited, is called an infinite series.

554. An infinite series is said to be convergent when the sum of the first n terms cannot numerically exceed some finite quantity however great n may be.

555. An infinite series is said to be divergent when the sum of the first n terms can be made numerically greater than any finite quantity, by taking n large enough.

556. By the sum of an infinite series is meant the limit towards which we approximate by continually adding more and more of its terms.

For example, consider the infinite series

Hence if x be less than 1, however great n may be, the sum

1

of the first n terms of the series is less than ; the series is 1 XC

therefore convergent. And as by taking n large enough, the sum

of the first n terms can be made to differ from

1

a quantity as we please,

is the sum of the infinite series.

1

Ꮳ

If x=1, the series is divergent; for the sum of the first n terms is n, and by taking sufficient terms this may be made greater than any finite quantity.

any finite quantity by taking n large enough.

557. An infinite series in which all the terms are of the same

sign is divergent if each term is greater than some assigned finite quantity, however small.

T. A.

22

For if each term is greater than the quantity c, the sum of the first n terms is greater than nc, and this can be made greater than any finite quantity by taking n large enough.

558. An infinite series of terms, the signs of which are alternately positive and negative, is convergent if each term be numerically less than the preceding term.

......

From the first mode of writing the series we see that the sum of any number of terms is a positive quantity, and from the second mode of writing the series we see that the sum of any number of terms is less than u1; hence the series is convergent.

It is necessary to shew in this case that the sum of any number of terms is positive; because if we only know that the sum is less than u1, we are not certain that it is not a negative quantity of unlimited magnitude.

559. An infinite series is convergent if from and after any fixed term the ratio of each term to the preceding term is numerically less than some quantity which is itself numerically less than unity.

Let the series beginning at the fixed term be

and let S denote the sum of the first n of these terms. Then

Now first let all the terms be positive, and suppose

Then S is less than u, {1+ k + k2 + ......+k" }; that is, less Hence if k be less than unity, S is less than

1-k"

than 1-k

U1 1

k

; thus the sum of as many terms as we please beginning

with U1 is less than a certain finite quantity, and therefore the series beginning with u, is convergent.

Secondly, suppose the terms not all positive; then if they are all negative, the numerical value of the sum of any number of them is the same as if they were all positive; if some are positive and some negative, the sum is numerically less than if they were all positive. Hence the infinite series is still convergent.

Since the infinite series beginning with u, is convergent, the infinite series which begins with any fixed term before u, will be also convergent; for we shall thus only have to add a finite number of finite terms to the series beginning with u,.

560. An infinite series is divergent if from and after any fixed term the ratio of each term to the preceding term is greater than unity, or equal to unity, and the terms are all of the same sign.

Let the series beginning at the fixed term be

and let S denote the sum of the first n of these terms. Then

u, {1+1+

Then S is numerically greater than

1

+ 1},

that is, numerically greater than nu ̧. Hence S may be made numerically greater than any finite quantity by taking n large enough, and therefore the series beginning with u, is divergent.