Algebra

Convergency and Divergency of Series

Let 𝑎₁+𝑎₂+𝑎₃+⋯ be a series, and 𝑎_𝑛, 𝑎_𝑛+1 any two consecutive terms. The following tests of convergency may be applied. The following test of convergency may be applied. The series will converge, if, after any fixed term:

1. The terms decrease and are alternately positive and negative.
2. Or if 𝑎_𝑛𝑎_𝑛+1 is always greater than some quantity greater than unity
3. Or if 𝑎_𝑛𝑎_𝑛+1 is never less than the corresponding ratio in a known converging series.
4. Or if 𝑛𝑎_𝑛𝑎_𝑛+1−𝑛 is always greater than some quantity greater than unity. By (244) and rule 3
5. Or if 𝑛𝑎_𝑛𝑎_𝑛+1−𝑛−1log𝑛 is always greater than some quantity greater than unity.

239 The conditions of divergency are obviously the converse of rules 1 to 3. 240 The series 𝑎₁+𝑎₂+𝑎₃+⋯ converges, if 𝑎_𝑛+1𝑎_𝑛 is always less than some quantity 𝑝, and 𝑥 less than 1𝑝. By {239) rule 2.241 To make the sum of the last series less than an assigned quantity 𝑝, make 𝑥 less than 𝑝𝑝+𝑘, 𝑘 being the greatest coefficient.242

General Theorem

If 𝜙(𝑥) be positive for all positive integral values of 𝑥, and continually diminish as 𝑥 increases, and if 𝑚 be any positive integer, then the two series 𝜙(1)+𝜙(2)+𝜙(3)+𝜙(4)+⋯ 𝜙(1)+𝑚𝜙(𝑚)+𝑚²𝜙(𝑚²)+𝑚³𝜙(𝑚³)+⋯ are either both convergent or divergent.243 Application of this theorem. To ascertain whether the series 11^𝑝+12^𝑝+13^𝑝+14^𝑝+⋯ is divergent or convergent when 𝑝 is greater than unity. Taking 𝑚=2, the second series in (243) becomes 1+22^𝑝+44^𝑝+88^𝑝+⋯ a geometrical progression which converges; therefore the given series converges.244 The series of which 1𝑛(log𝑛)^𝑝 is the general term is convergent if 𝑝 be greater than unity, and divergent if 𝑝 be not greater than unity. By (243), (244). 245 The series of which the general term is 1𝑛𝜆(𝑛)𝜆²(𝑛)⋯𝜆^𝑟(𝑛){𝜆^𝑟+1(𝑛)}^𝑝 where 𝜆(𝑛) signifies log 𝑛, 𝜆²(𝑛) signifies log{log(𝑛)}, and so on, is convergent if 𝑝 be greater than unity, and divergent if 𝑝 be not greater than unity. By induction, and by (243).246 The series 𝑎₁+𝑎₂+𝑎₃+⋯ is convergent if 𝑛𝑎_𝑛log(n)log²(n)⋯log^𝑟(n){log_𝑟+1(𝑛)}^𝑝 is always finite for a value of 𝑝 greater than unity; log²(n) here signifying log(log n), and so on. See Tedhuter's Algebra or Boole's Finite Differences.246

Sources and References

https://archive.org/details/synopsis-of-elementary-results-in-pure-and-applied-mathematics-pdfdrive