ofdawly converging and diverging Injimte Series, S5l 



In the series above marked (B.) it is plain that if we stop 

 at the term involving A", we shall obtain an inferior limit to 

 the Slim S, and if we take in the term immediately following 

 we shall get a superior limit. 



Now, to narrow these limits, we must observe, that since 



A«+i= A"- A", 



a superior limit will be obtained by adding to the inferior 

 limit the quantity 



Moreover, the inferior limit cannot differ from the entire sum 

 by so much as 



whereas the superior limit differs from the truth by more than 



A"'^+^ 

 (<r+l)«+i' 



in as much as the succeeding term in the series (B.) is also 

 negative. The inferior limit will therefore be nearer to the 

 truth than the superior, since A"'>A"+^and, consequently, 

 half the sum of the two limits will be a superior limit still 

 nearer. We may conclude, therefore, that if we multiply the 



final term in the inferior limit by — , and add the product to 



that limit, we shall thus obtain a near superior limit. 



It is scarcely necessary to observe here, that in what has 

 been hitherto said, the coefficients in the proposed series S 

 are supposed continually to diminish, as also the several series 

 of differences, which supposition is conformable to what 

 usually occurs in practice when S is convergent. When the 

 series is divergent, the formula (B.) or (A.) is still applicable, 

 regard being paid to the signs of the differences. 



1. As a first example, let there be proposed the slowly con- 

 verging series 



which expresses the length of the quadrantal arc of a circle 

 whose diameter is 1. It will be advisable to actually sum up 

 a few of the leading terms, and to apply the formula (B.), or 

 rather (C), to the remaining part. The work may be ar- 

 ranged as follows : 



