312 ALGEBRA.
make the fraction 1 2 n-1 as small as we please. Thus by taking a sufficient number of terms the sum can be made to differ by as little as we please from 2.
In the next article a more general case is discussed.
381. Sum to Infinity. From Art. 379 we have S = a(1 — r n) 1 — r= a 1 — r - arn 1 — r
Suppose r is a proper fraction; then the greater the value of n the smaller is the value of rn, and consequently of a rn 1-r ; and therefore by making n sufficiently large, we can make the sum of n terms of the series differ from a 1-r by as small a quantity as we please.
This result is usually stated thus : the sum of an infinite number of terms of a decreasing Geometrical Progression is a 1-r or more briefly, the sum to infinity is a 1-r
382. Recurring decimals furnish a good illustration of Infinite Geometrical Progressions.
Ex. Find the value of .423.
.423 = .4232323 ... = 4 10 + 23 1000 + 23 100000 + ... = 4 10 + 23 103 + 23 105 4 10 ion 102 104 I 10 10^ 23 1 10^ 1 _ J_ 4 , 23 100 4 , 23 419 102 10 "^ 103 '99 10 "^ 990 990'
which agrees with the value found by the usual arithmetical rule.
EXAMPLES XXXIV. d.
Sum to infinity the following series:
1. 9,6,4,... 2. 12,6,3,... 3. 12, 14, 18,... 4. 12, -14, 18,... 5. 13,29,4 27,... 6. 85, -1, 58,... 7. .9, .03, .001,... 8. .8, -.4, .2, .
Find by the method of Art. 382, the value of
9. .3. 10. .16. 11. .24. 12. .378. 13. .037.
