Series

Such tests as exist for series of this type, which are not ab solutely convergent, depend upon a theorem known as Abel's Lemma, which is that, if a3. • • • is a steadily decreasing set of numbers, then • • • -+„u„ is less than and greater than where A and B are respectively the great est and least of the expressions It follows that, if is convergent and is steadily increasing or steadily decreasing to a finite limit, then is convergent. It follows also that, if I ui-F-u2+ • • is always less than some fixed number independent of n, and decreases steadily to zero, then is a convergent series. A particular case is that, if decreases steadily to zero, the series cos (n0 + a) is convergent. A still more particular case is that the series • • • is convergent. Such a series is called an alternating series.

If lv„ is a convergent series and tends to a limit 1, different from zero, it cannot be inferred, if and are not always of the same sign, that is also convergent. It may however happen that the convergence or divergence of the series can be readily established. If this series is con vergent, then is convergent. If it is not convergent, then is not convergent. For example, if v„= ( - then is convergent, if p is positive. If u„= (- {TIP+ ( then Now v„- u„ = {nP+(- , which is always positive and is convergent or divergent with the series and so is convergent only if 2p> 1. Hence Lu„ is convergent only if 2p>I.

Series of Complex Terms.

A series of complex terms I(a„-Fib„) can only be convergent if and Ib„ are separately convergent. If 1 is convergent, the series is said to be absolutely convergent. In this case and Ib„ are each ab solutely convergent. and the converse also holds. Certain tests can be extended to series of complex terms. Thus by an extension of Maclaurin's Integral test the series where A=a+ig, may be shown to be convergent only if a > 1. The following is an extension of the ratio-test. If I k> 1, then the series is absolutely convergent if the real part of ,u is greater than I and is not convergent at all in any other case.

Double Series.

A double series is formed by the addition of terms a„„„ depending on two integers. We may suppose the term to occupy the point whose cartesian co-ordinates re ferred to two axes are m,n. The sum of all the terms inside the rectangle bounded by the axes and the lines x = m+i, y = n+ may be denoted by The double series may be said to con verge if tends to a definite limit as m and n each tend inde pendently to infinity. Other methods of summation are as follows. Keeping n fixed we may try to sum the infinite series an,' an,2+ ' • • • If this converges to a sum A„, we may then sum the infinite series A +A 2+ • . If this series converges we call the sum the

sum by rows of the double series. We can similarly investigate a sum by columns. But it does not follow that, if the double series has a sum in the first sense, it has a sum either by rows or columns equal to this sum. This does actually happen when the rows and columns are both convergent and the double series is convergent in the first sense. If the double series is not convergent in the first sense, the sums by rows and columns may both exist but are not necessarily equal, nor does the fact that the sums by rows and columns both exist and are equal imply that the double series is convergent in the first sense.

Another method of summation is to form a single series Ic„, where cm = + • • • -Panto. If this series is convergent, its sum is called the sum of the double series by diagonals. The existence of this sum does not imply the existence of any other sum. If all the terms of a double series are positive, then, if it can be summed in any way, it can be summed in every other way and all the sums are the same, and are unaffected by any derange ment of its terms. The idea of absolute convergence may be extended to double series. Tests for double series of positive terms may be obtained analogous to those for single series. Thus, if f(x, y) is positive and steadily decreases to zero as x and y in crease, the double series l'Ef(ni, n) and the double integral f'f'f(x, y)dx dy are convergent or divergent together. Thus the double series is convergent if a> I, but di vergent otherwise, a result having an application of fundamental importance in the theory of elliptic functions. Triple and mul tiple series are obvious extensions of double series.

Multiplication of Series.

The most interesting case of double series is perhaps that furnished by the multiplication of series. We are able to exhibit the product of the series and Ev„ as a double series If /u„ converges to a sum U and converges to a sum V, then since is equal to the double series is always convergent in the first sense. The sum usually taken as the product of the two series is however the sum of the double series by diagonals, that is, Zw„, where This arises from the fact that the formal product of the two series u„xn , arranged in powers of x, is Zw„xn. The chief theo rems concerning this product, sometimes specifically alluded to as Cauchy's product, are (I) Cauchy's theorem. If and converge absolutely to the sums U and V respectively, then 'w,, converges absolutely to UV.