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INFINITE SERIES.

ON SERIES.

We know that when x is less than 1, the series

1 + x + x2 + ...

81

is of such a nature that the sum of the first n terms can

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enough. For the sum of the first ʼn terms is

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since x is less than 1, x" can be made as small as we like by taking n large enough. The value to which the sum of the first n terms of a series can be made to approach as near as we like by making n large enough is called the sum of the series. It should be observed that the word sum is here used in a new sense, and we must not assume without proof that what is true of the old sense is true of the new one: e.g. that the sum is independent of the order of the terms. When a series has a sum it is said to be convergent. When the sum of n terms can be made to exceed any proposed quantity in absolute value by taking n large enough, the series is called divergent.

A series whose terms are all positive is convergent if there is a positive quantity which the sum of the first n terms never surpasses, however large n may be. For consider two quantities, one which the sum surpasses, and one which it does not. All quantities between these two must fall into two groups, those which the sum surpasses when n is taken large enough, and those which it does not. These groups must be separated from one another by a single quantity which is the least of those which the sum does not surpass; for there can be no quantities between the two groups. This single quantity has the property that the sum of the first n terms can be brought as near to it as we please, for it can be made to surpass every less quantity.

The same thing holds when all the terms are negative, if there is a negative quantity which the sum of the first n terms never surpasses in absolute magnitude.

When the terms are all of the same sign, the sum of the series is independent of the order of the terms. For let P be the sum of the first n terms and P the sum of the series, when the terms are arranged in one order; and let Q be the sum of the first n terms and Q the sum of the series, when the terms are arranged in another order. Then P cannot exceed Q, nor can Q exceed P; and P„, Q can be brought as near as we like to P, Q by taking n large enough. Hence P cannot exceed Q, nor can exceed P; that is, P=Q.

n

n

m

When the terms are of different signs, we may separate the series into two, one consisting of the positive terms and the other of the negative terms. If one of these is divergent and not the other, it is clear that the combined series is divergent. If both are convergent, the combined series has a sum independent of the order of the terms. For let P be the sum of m terms of the positive series, -Q, the sum of n terms of the negative series, P, Q, the sums of the two series respectively; and suppose that in the first m+n terms of the compound series there are m positive and n negative terms, so that the sum of those m+n terms is Pm Qn. Then P-Pm, Q-Qn can be. made as small as we like by taking m, n large enough; therefore P-Q-(P- Q) can be made as small as we like by taking m+n large enough, or P-Q is the sum of the compound series. It is here assumed that by taking sufficient terms of the compound series we can get as many positive and as many negative terms as we like. If, for example, we could not get as many negative terms as we liked, there would be a finite number of negative terms mixed up with an infinite series of positive terms, and the sum would of course be independent of the order.

If, however, the positive and negative series are both divergent, while the terms in each of them diminish without limit as we advance in the series, it is possible to make the sum of the compound series equal to any arbitrary quantity C by taking the terms in a suitable order. Suppose C positive; take enough positive terms to bring their sum above C, then enough negative terms to bring the sum below C, then enough positive terms to bring the sum again above C, and so on. We can always per

EXPONENTIAL SERIES.

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form each of these operations, because each of the series is divergent; and the sum of n terms of the compound series so formed can be made to differ from C as little as we like by taking n large enough, because the terms decrease without limit.

Putting these results together, we may say that thé sum of a series is independent of the order of the terms if, and only if, the series converges when we make all the terms positive.

EXPONENTIAL SERIES.

We shall now find a series for e, which is the result of making unity grow at the log. rate 1 for x seconds. Suppose that

...

ex = a + bx + cx2 + dx3 + that is, suppose it is possible to find a, b, c... so that the series shall be convergent and have the sum e*. We will assume also (what will have to be proved) that the flux of the sum of the series is itself the sum of a series whose terms are the fluxes of the terms of the original series. Now the flux of e" is e*, because it grows at the logarithmic rate 1. Hence we have

e=b+2cx + 3dx2 + ...

and this must be the same series as before. Hence

b=a, 2c=b, 3d = c, etc.

Now by putting x=

O we see that a = 1, because e° is the result of making unity grow for no time. Writing then for shortness IIn instead of 1.2.3.

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n, we find

e"=1+x+ + + + ... + +... f(x), say.

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This is called the exponential series. We shall now verify this result by an accurate investigation.

The exponential series is convergent for all values of x. For taken larger than x; then the series after the nth term may be written thus:

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and each term after the first two of the quantity in the brackets is less than the corresponding term of

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n+1 (n + 1)2 + .

...

which is convergent. And since it is convergent when the terms are all positive, the sum is independent of the order of the terms.

The sum of the exponential series increases at log. rate 1. Consider four quantities, x, x,, x, x, in ascending order of magnitude. We find for the mean flux from x, to x2, x1 M _f (x) − f(x) X, +∞2 + x2+x‚ׂ« + x2 + ...

=

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2

x2

+

2

x,

= 1 +

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2

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(Observe that the order of the terms has been changed, and why this is lawful.) Each term of this series is less than the corresponding term of f(x), and greater than the corresponding term of f(x). Hence the series is convergent, and its sum M lies between f(x) and ƒ(x). And since M is finite,

M(x,-x) or f(x1) — ƒ (x),

can be made as small as we like by making x1— æ, small enough. Hence also f(x) -ƒ (x) can be made as small as we like by making xx, small enough. Consequently we can find an interval (from x to x) such that the mean flux M of every included interval (from x, to x) differs from f(x) less than by a proposed quantity, however small. Therefore f(x) is the flux of f(x), or the sum of the exponential series increases at log. rate 1.

It follows that f(x) = e*; for both quantities increase at the log. rate 1, and they are equal when x = 0, therefore always equal.

It appears from the investigation above, that if ƒ (x) denote the sum of a convergent series proceeding by powers of x, and f'(x) the sum of the derived series got by taking the flux of every term; then f'(x) will be the flux of f(x) whenever ƒ (x) —ƒ'(y) can be made

COMPLEX NUMBERS,

85

as small as we like by taking a -y small enough; that is, when f'x varies continuously in the neighbourhood of the value x.

By putting x1, we find the value of the quantity e; it is 2-718281828...

THE LOGARITHMIC SPIRAL.

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We may convert a step oa into a step ob by turning it through the angle aob and altering its length in the ratio oa ob. But this operation may be divided into two simpler parts. From b draw bm perpendicular to oa, then ob = om + mb. Now we may convert oa into om by simply increasing its length in the ratio oa: om. Let om so that om = x. oα. drawn perpendicular to oa, and equal to it in length, we can convert oa' into mb by multiplying it by a numerical ratio y, such that mb=y. oa'. Now we can convert oa into oa by turning it counter-clockwise through a right angle. Let i denote this operation; then

Consequently
And finally

ob

oα = x,

If oa' is

oa' = i.oa.

mb = y. oa' = yi. oa.

= om + mb

α m

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= x.oa + yi, oa = (x + yi) oa. Thus the operation which converts oa into ob may be written in the form x+yi, where x and y are numerical ratios, and i is the operation of turning counter-clockwise through a right angle. This meaning is quite different from that which we formerly gave to the letter i. We shall never use the two meanings at the same time, in speaking of steps in one plane.

If oa be taken of the unit length, every other step ob in the plane may be represented by means of its ratio to this unit; for oa being = 1,

ob = (x + yi) oα = x + yi.

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