An elementary treatise on the differential calculus, in which the method of limits is exclusively made use of, by the Rev. M. O'Brien.

THE REMAINDER IN TAYLOR S SERIES. 85
1 and 1 + - are evidently 0 and r4 { - 1). Hence
10 (1+ 1 5 '
the error lies between 0 and - - and we have
5 (1ll 105/
therefore
/ 1 1 1 1 1 1 1 1 1 1 1
log I +- _ + _ -_
=10 10 2 10" 3 - 4 104 5 105
~1 1 \
+ an error between o and ( -
5 1 15 105'
Since =.000006... this error lies between 0 and.000004.
-. 0 -which has no significant digit in the first 6
5
decimal places.
Ex.2. Let f(x)=a, a=l, h =-, n=5; then
f (n) = l ( - 1) ( ) (L - n + 1) v-a;
1.3.5
the greatest and least values of this for all values of x between
1 and 1 + - are 0 and
10
1.3.57 i o 7
25 + -~
Hence the error lies between 0 and
1.3.5.7 f 1 y 1
25r5 IF 5 Io9 T
o7 ( 1 1 —
7 i 1 /i ll'
or - - -..
156 105 115 10