KXPONENTIAL AND LOGARITHMIC SERIES 13 



If the pendulum actually beats seconds at the lower tempera- 

 ture, then we can put n x = 1 in the relation which gives the loss 

 per second. 



9. The Ea-poneniial and Logarithmic Series. 

 Putting a =- in the expansion 



(l + a)-l + n a+ ^I>' + n( "-| ) 8 (re - 2) q3 + . . . 



we get 



.n(n-l)(n-2)^V . 



13 ~\nJ + ' ' ' 



Making n infinitely great, all of the fractions having n for their 

 denominators become infinitely small and can therefore be 

 neglected. Thus in the limit the right hand side becomes 



a quantity having a definite value, calling this quantity e 

 then , =1+1+ A_ + _!.+ ... 



By evaluating the series the value of e can be found correct to 

 as many significant figures as required. 



To find e correct to six significant figures. 



1-000000 

 21-000000 

 500000 

 -166667 

 -041667 

 008333 

 -001389 

 -000198 

 000025 

 -000003 



2-718282 

 e = 2-71828 correct to six significant figures. 



(l\n (/ l\ n \ x 

 1+-) =|M+-)| =6* when n is made infinitely 



