Another very important expansion in powers of is that for . Since and if is numerically less than unity, it is natural to expect that will be equal, when , to the series obtained by integrating each term of the series from to , to the series . And this is in fact the case. For and so, if , where
We require to show that the limit of , when tends to , is zero. This is almost obvious when ; for then is positive and less than and therefore less than . If on the other hand , we put and , so that which shows that has the sign of . Also, since the greatest value of in the range of integration is , we have and so .
Hence provided that . If lies outside these limits the series is not convergent. If we obtain a result already proved otherwise (Ex. LXXXIX. 7).