212 Algebra. 



This number is one of the most important constants in mathe- 

 matics. It is called the Naperia?i Base and is always represented 

 by the letter e. Its value is known to more than 260 decimal 

 places. 



30. Logarithmic Series. The Logarithmic Series is the ex- 

 pansion o{\oge(i-\-x) in terms of the ascending powers of .r. 

 From the exponential series 



Whence, transposing the i and dividing through by y, 



y ~ ' ( h 1 3 ' ' ' \ 



Therefore, since these variables are always equal, ^ 



limit (^^' — II limit L , r(loge«)' ,J<logea)' , 1 ") . . 



Whence it is easy to see 



limit {a- 



Ao'gea. (4) 



Now put I -f-.v for a, then we have 



. , , limit ((1+X/—1) 



iogeri+^r;=^ ;^ 4 — ^- — i 



Expanding (i-{-xy by binomial formula, 



ios,.ri+-v;=;;""M-''-.v»+fc!i^^-43 



r ^ o( 1 . 2 1.2.3 



1.2.3.4 ) 



The limit of the right member as r ^ o can be plainly seen ; 

 whence we obtain the equation 



iog,rn-x;=x-^V^"-"^-+ ... (6) 



234 



This is the LogaritliDiic Series. 



31. Convergency of the Series. The above seri,f;s is not 

 convergent for values of x greater than i , and hence cannot be 

 used for computing the logarithm of any integral number but 2. 

 The following scheme will give a series which is available for 

 computing the logarithms of all integers. 



