76. Infinite Series.
Suppose that \(u(n)\) is any function of \(n\) defined for all values of \(n\). If we add up the values of \(u(\nu)\) for \(\nu = 1\), \(2\), … \(n\), we obtain another function of \(n\), viz. \[s(n) = u(1) + u(2) + \dots + u(n),\] also defined for all values of \(n\). It is generally most convenient to alter our notation slightly and write this equation in the form \[s_{n} = u_{1} + u_{2} + \dots + u_{n},\] or, more shortly, \[s_{n} = \sum_{\nu=1}^{n} u_{\nu}.\]
If now we suppose that \(s_{n}\) tends to a limit \(s\) when \(n\) tends to \(\infty\), we have \[\lim_{n\to\infty} \sum_{\nu=1}^{n} u_{\nu} = s.\] This equation is usually written in one of the forms \[\sum_{\nu=1}^{\infty} u_{\nu} = s,\quad u_{1} + u_{2} + u_{3} + \dots = s,\] the dots denoting the indefinite continuance of the series of \(u\)’s.
The meaning of the above equations, expressed roughly, is that by adding more and more of the \(u\)’s together we get nearer and nearer to the limit \(s\). More precisely, if any small positive number \(\epsilon\) is chosen, we can choose \(n_{0}(\epsilon)\) so that the sum of the first \(n_{0}(\epsilon)\) terms, or any of greater number of terms, lies between \(s – \epsilon\) and \(s + \epsilon\); or in symbols \[s – \epsilon < s_{n} < s + \epsilon,\] if \(n \geq n_{0}(\epsilon)\). In these circumstances we shall call the series \[u_{1} + u_{2} + \dots\] a convergent infinite series, and we shall call \(s\) the sum of the series, or the sum of all the terms of the series.
Thus to say that the series \(u_{1} + u_{2} + \dots\) converges and has the sum \(s\), or converges to the sum \(s\) or simply converges to \(s\), is merely another way of stating that the sum \(s_{n} = u_{1} + u_{2} + \dots + u_{n}\) of the first \(n\) terms tends to the limit \(s\) as \(n \to \infty\), and the consideration of such infinite series introduces no new ideas beyond those with which the early part of this chapter should already have made the reader familiar. In fact the sum \(s_{n}\) is merely a function \(\phi(n)\), such as we have been considering, expressed in a particular form. Any function \(\phi(n)\) may be expressed in this form, by writing \[\phi(n) = \phi(1) + \{\phi(2) – \phi(1)\} + \dots + \{\phi(n) – \phi(n – 1)\};\] and it is sometimes convenient to say that \(\phi(n)\) converges (instead of ‘tends’) to the limit \(l\), say, as \(n \to \infty\).
If \(s_{n} \to +\infty\) or \(s_{n} \to -\infty\), we shall say that the series \(u_{1} + u_{2} + \dots\) is divergent or diverges to \(+\infty\), or \(-\infty\), as the case may be. These phrases too may be applied to any function \(\phi(n)\): thus if \(\phi(n) \to +\infty\) we may say that \(\phi(n)\) diverges to \(+\infty\). If \(s_{n}\) does not tend to a limit or to \(+\infty\) or to \(-\infty\), then it oscillates finitely or infinitely: in this case we say that the series \(u_{1} + u_{2} + \dots\) oscillates finitely or infinitely.1
77. General theorems concerning infinite series.
When we are dealing with infinite series we shall constantly have occasion to use the following general theorems.
(1) If \(u_{1} + u_{2} + \dots\) is convergent, and has the sum \(s\), then \(a + u_{1} + u_{2} + \dots\) is convergent and has the sum \(a + s\). Similarly \(a + b + c + \dots + k + u_{1} + u_{2} + \dots\) is convergent and has the sum \(a + b + c + \dots + k + s\).
(2) If \(u_{1} + u_{2} + \dots\) is convergent and has the sum \(s\), then \(u_{m+1} + u_{m+2} + \dots\) is convergent and has the sum \[s – u_{1} – u_{2} – \dots – u_{m}.\]
(3) If any series considered in (1) or (2) diverges or oscillates, then so do the others.
(4) If \(u_{1} + u_{2} + \dots\) is convergent and has the sum \(s\), then \(ku_{1} + ku_{2} + \dots\) is convergent and has the sum \(ks\).
(5) If the first series considered in (4) diverges or oscillates, then so does the second, unless \(k = 0\).
(6) If \(u_{1} + u_{2} + \dots\) and \(v_{1} + v_{2} + \dots\) are both convergent, then the series \((u_{1} + v_{1}) + (u_{2} + v_{2}) + \dots\) is convergent and its sum is the sum of the first two series.
All these theorems are almost obvious and may be proved at once from the definitions or by applying the results of §§ 63-66 to the sum \(s_{n} = u_{1} + u_{2} + \dots + u_{n}\). Those which follow are of a somewhat different character.
(7) If \(u_{1} + u_{2} + \dots\) is convergent, then \(\lim u_{n} = 0\).
For \(u_{n} = s_{n} – s_{n-1}\), and \(s_{n}\) and \(s_{n-1}\) have the same limit \(s\). Hence \(\lim u_{n} = s – s = 0\).
The reader may be tempted to think that the converse of the theorem is true and that if \(\lim u_{n} = 0\) then the series \(\sum u_{n}\) must be convergent. That this is not the case is easily seen from an example. Let the series be \[1 + \tfrac{1}{2} + \tfrac{1}{3} + \tfrac{1}{4} + \dots\] so that \(u_{n} = 1/n\). The sum of the first four terms is \[1 + \tfrac{1}{2} + \tfrac{1}{3} + \tfrac{1}{4} > 1 + \tfrac{1}{2} + \tfrac{2}{4} = 1 + \tfrac{1}{2} + \tfrac{1}{2}.\] The sum of the next four terms is \(\frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} > \frac{4}{8} = \frac{1}{2}\); the sum of the next eight terms is greater than \(\frac{8}{16} = \frac{1}{2}\), and so on. The sum of the first \[4 + 4 + 8 + 16 + \dots + 2^{n} = 2^{n+1}\] terms is greater than \[2 + \tfrac{1}{2} + \tfrac{1}{2} + \tfrac{1}{2} + \dots + \tfrac{1}{2} = \tfrac{1}{2} (n + 3),\] and this increases beyond all limit with \(n\): hence the series diverges to \(+\infty\).
8) If \(u_{1} + u_{2} + u_{3} + \dots\) is convergent, then so is any series formed by grouping the terms in brackets in any way to form new single terms, and the sums of the two series are the same.
The reader will be able to supply the proof of this theorem. Here again the converse is not true. Thus \(1 – 1 + 1 – 1 + \dots\) oscillates, while \[(1 – 1) + (1 – 1) + \dots\] or \(0 + 0 + 0 + \dots\) converges to \(0\).
(9) If every term \(u_{n}\) is positive \(or zero\), then the series \(\sum u_{n}\) must either converge or diverge to \(+\infty\). If it converges, its sum must be positive (unless all the terms are zero, when of course its sum is zero).
For \(s_{n}\) is an increasing function of \(n\), according to the definition of § 69, and we can apply the results of that section to \(s_{n}\).
(10) If every term \(u_{n}\) is positive \(or zero\), then the necessary and sufficient condition that the series \(\sum u_{n}\) should be convergent is that it should be possible to find a number \(K\) such that the sum of any number of terms is less than \(K\); and, if \(K\) can be so found, then the sum of the series is not greater than \(K\).
This also follows at once from § 69. It is perhaps hardly necessary to point out that the theorem is not true if the condition that every \(u_{n}\) is positive is not fulfilled. For example \[1 – 1 + 1 – 1 + \dots\] obviously oscillates, \(s_{n}\) being alternately equal to \(1\) and to \(0\).
(11) If \(u_{1} + u_{2} + \dots\), \(v_{1} + v_{2} + \dots\) are two series of positive \(or zero\) terms, and the second series is convergent, and if \(u_{n} \leq Kv_{n}\), where \(K\) is a constant, for all values of \(n\), then the first series is also convergent, and its sum is less than or equal to \(K\) times that of the second.
For if \(v_{1} + v_{2} + \dots = t\) then \(v_{1} + v_{2} + \dots + v_{n} \leq t\) for all values of \(n\), and so \(u_{1} + u_{2} + \dots + u_{n} \leq Kt\); which proves the theorem.
Conversely, if \(\sum u_{n}\) is divergent, and \(v_{n} \geq Ku_{n}\), then \(\sum v_{n}\) is divergent.
- The reader should be warned that the words ‘divergent’ and ‘oscillatory’ are used differently by different writers. The use of the words here agrees with that of Bromwich’s Infinite Series. In Hobson’s Theory of Functions of a Real Variable a series is said to oscillate only if it oscillates finitely, series which oscillate infinitely being classed as ‘divergent’. Many foreign writers use ‘divergent’ as meaning merely ‘not convergent’.↩︎
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