A more general test, which includes the test of § 188 as a particular test case, is the following.

**Dirichlet’s Test.** If \(\phi_{n}\) satisfies the same conditions as in § 188, and \(\sum a_{n}\) is any series which converges or oscillates finitely, then the series \[a_{0}\phi_{0} + a_{1}\phi_{1} + a_{2}\phi_{2} + \dots\] is convergent.

The reader will easily verify the identity \[a_{0}\phi_{0} + a_{1}\phi_{1} + \dots + a_{n}\phi_{n} = s_{0}(\phi_{0} – \phi_{1}) + s_{1}(\phi_{1} – \phi_{2}) + \dots + s_{n-1}(\phi_{n-1} – \phi_{n}) + s_{n}\phi_{n},\] where \(s_{n} = a_{0} + a_{1} + \dots + a_{n}\). Now the series \((\phi_{0} – \phi_{1}) + (\phi_{1} – \phi_{2}) + \dots\) is convergent, since the sum to \(n\) terms is \(\phi_{0} – \phi_{n}\) and \(\lim \phi_{n} = 0\); and all its terms are positive. Also since \(\sum a_{n}\), if not actually convergent, at any rate oscillates finitely, we can determine a constant \(K\) so that \(|s_{\nu}| < K\) for all values of \(\nu\). Hence the series \[\sum s_{\nu}(\phi_{\nu} – \phi_{\nu+1})\] is absolutely convergent, and so \[s_{0}(\phi_{0} – \phi_{1}) + s_{1}(\phi_{1} – \phi_{2}) + \dots + s_{n-1}(\phi_{n-1} – \phi_{n})\] tends to a limit as \(n \to \infty\). Also \(\phi_{n}\), and therefore \(s_{n}\phi_{n}\), tends to zero. And therefore \[a_{0}\phi_{0} + a_{1}\phi_{1} + \dots + a_{n}\phi_{n}\] tends to a limit, the series \(\sum a_{\nu}\phi_{\nu}\) is convergent.

**Abel’s Test.** There is another test, due to Abel, which, though of less frequent application than Dirichlet’s, is sometimes useful.

Suppose that \(\phi_{n}\), as in Dirichlet’s Test, is a positive and decreasing function of \(n\), but that its limit as \(n \to \infty\) is not necessarily zero. Thus we postulate less about \(\phi_{n}\), but to make up for this we postulate more about \(\sum a_{n}\), viz. that it is *convergent*. Then we have the theorem:

if \(\phi_{n}\) is a positive and decreasing function of \(n\), and \(\sum a_{n}\) is convergent, then \(\sum a_{n}\phi_{n}\) is convergent.

For \(\phi_{n}\) has a limit as \(n \to \infty\), say \(l\): and \(\lim (\phi_{n} – l) = 0\). Hence, by Dirichlet’s Test, \(\sum a_{n}(\phi_{n} – l)\) is convergent; and as \(\sum{a_{n}}\) is convergent it follows that \(\sum a_{n}\phi_{n}\) is convergent.

This theorem may be stated as follows:

*a convergent series remains convergent if we multiply its terms by any sequence of positive and decreasing factors.*

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