41 6 FROM ELEMENTARY ALGEIJRA TO THE CALCULUS. 



Theorems which have to be buttressed by theorems on ' Con- 

 vergence ' — ■ 



[Of course, the one fundamental test of convergence, that w„/«,._, 



- a f -+ . . where either a < i, or a =i and 6 > i, is necessary 



n 

 as a test of the intelhgibihty of any infinite series that may 

 arise : but it is unnecessary and undesirable to nterpose some- 

 what amorphous Higher Algebra between simple Algebra and 

 the Calculus. The trend of modern mathematics is to jett son 

 this Higher Algebra; and the one object of this paper is to 

 attempt to show how this can he done with not loss but gain 

 of security and completeness in our foundations, and so to 

 show how the Calculus can be made readily accessible to the 

 average Cape Intermediate student]. 



Since /(a) - f{o) = I f\x) dx 



(changing .v to x - z). 



I 



f'{x-z)dz. 

 Integrating by parts, -xfo^j z. j"\x-z)dz. 



= xf'o-\- I f'\x-z).dz- 

 J o ' 2 



.-\nd repeating this process, 



fix) -fo^-xf'o-\-%f"{o)+ .... -l-g' /-\ 



/-'' z' 



/■''{x - z).d^ 



with the conditions, involved in our proof of the Summation 

 Theorem, that /(v), f'{x) and the higher derivatives are con- 

 tinuous from to x of the variable. 



Thus Maclaurin's Theorem holds as an infinite expansion 



/-^ z'' 



^{x - z) ^_ tends to o as N — >- oo 



Lagrange's remainder follows easily, and is not, it seems to 

 me, worth proving in any other way. 



But, as is well known, Lagrange's remainder fails to settle 

 the convergence of the Binomial and logarithmic series. 



These are settled by means of a simple change of variable 

 in the above Integral. The processes are hardly worth our time 

 in such a gathering as this — but were too much for Honours 

 students last year. I will conclude m\- paper with two more 

 abstruse remarks : — 



(i) Differential coefficients versus Differentials. Differentials {i.e., 

 bare dx, dy) can be used in an expression whenever the 

 omitted denominator can be regarded as implied — e. 



''•i'- 



