74 JAJVIES BERNOULLI. 



128, James Bernoulli enforces tlie importance of the subject 

 in the following terms, page 243, 



Cseterum quantse sit necessitatis pariter et utilitatis hasc serierum 

 contemplatio, ei sane ignotum esse non poterit, qui perspectum habuerit, 

 ejusmodi series sacram quasi esse anchoram, ad quam in maxime arduis 

 et desperatse solutionis Problematibus, ubi omnes alias humani ingenii 

 vires naufragium passae, velut ultimi remedii loco confugiendum est. 



129. The principal artifice employed by James Bernoulli in 

 this memoir is that of subtracting one series from another, thus 

 obtaining a third series. For example, 



let /S'=l + R+iT+ ... + 



2 ' 3 n + l ' 



a ..11 11 



then b= l + -^ + o+"- + ~-^ TT 5 



z 3 n 71 + 1 



1 r ^ -, 111 11 



therefore = — 1 + ^ — ^ + ^ — ^ + - — - + . . . + -7 — — rr + 



1 . 2 ' 2 . 3 ' 3 . 4 ' •" ' 7i(?i + l) n + 1 ' 



, . Ill 1,1 



therelore -z — ^ + - — ^ + ^ — r + • • . H — 7 — — ty = 1 — 



1.2' 2. 33. 4' ' n{n+l) n+1' 



Thus the sum of n terms of the series, of which the r^^ term is 

 1 . n 



IS 



r (r + 1) ' n + 1 ' 



ISO. James Bernoulli says that his brother first observed 



1111 



that the sum of the infinite series -+ — +- + y + ...is infinite ; 



i. jLi O ^ 



and he gives his brother's demonstration and his own ; see his 

 page 250. 



131. James Bernoulli shews that the sum of the infinite series 

 _ _|_ — ^ + -j- . . . is finite, but confesses himself unable to give 



the sum. He says, page 254, Si quis inveniat nobisque commu- 

 nicet, quod industriam nostram elusit hactenus, magnas de nobis 



crratias feret. The sum is now known to be 7r ; this result is due 



to Euler : it is given in his Introductio in Analysin Infinitorum, 

 1748, Vol. L page 130. 



