n-1 + Un и n-1 which is easily seen to be equal to Un +1 un + u n+1 (i) will be equal to the (n + 1)th convergent of (ii) provided the sum of n terms of (i) is equal to the nth convergent of (ii). But it is easily seen that the theorem is true when n is 1 or 2 or 3: it is therefore true for all values of n. Thus u1 + u2+ug + u1 +...to n terms U1 = = Из 1 - U1 + Uz Uz + Uz Uz + Us It can be proved in a precisely similar manner that The formula [B] can also be deduced from [A] by changing the signs of the alternate terms. all the upper signs, or all the lower signs, being taken. all the upper signs, or all the lower signs, being taken. These can be proved by induction as in the preceding Article. and Thus to prove [C]. It is obvious by inspection that the theorem is true when n=2. Assume then that [C] is true for any particular value of n; then, to include another term of the series an must be will become b n±a n = b n + 1±an+1 Thus, if [C] be true for any value of n, it will be true for the next greater value; hence as [C] is true when n=2, it is true for all values of n. The following are particular cases of [C]. 1 1 12 32 52 1 + 2 + 2 + 2 + 1 + to infinity. [Brouncker.] ... [F].* to infinity=log,2. [Euler.] Put a1 =1, a2=2, a3=3, &c. in [D]. * The formula [A] is due to Euler; [C] is given by Dr Glaisher in the Proceedings of the London Mathematical Society, Vol. v. |