Determine wheather each of the following series is convergent. If so, find the sum of series. If not find the smallest values of n for which the sum of the first n terms of the series exceeds 100.
a) 18-9+4.5-...
b) 1.2+1.8+2.7+...
PLEASE HELP
and please in the long way explained..
+33653 "Determine wheather each of the following series is convergent. If so, find the sum of series. If not find the smallest values of n for which the sum of the first n terms of the series exceeds 100.
a) 18-9+4.5-...
b) 1.2+1.8+2.7+...
PLEASE HELP
and please in the long way explained.."
.
+23251 The first series: 18 - 9 + 4.5 - ...
appears to be a geometric series with common ratio -1/2.
Since the common ratio is a fraction between -1 and 1, there is an infinite sum:
Sum = a / (1 - r) = 18 / (1 - -1/2) = 18 / ( 1 + 1/2) = 18 / (3/2) = 12.
The second series: 1.2 + 1.8 + 2.7 + ...
appears to be a geometric series with common ratio 1.5.
Since the common ratio is not a fraction between -1 and 1, there is no infinite sum.
To find when the finite sum exceeds 100, use the formula: Sum = a(1 - rn) / (1 - r):
100 = 1.2(1 - 1.5n) / (1 - 1.5)
100 = 1.2(1 - 1.5n) / (-0.5)
-50 = 1.2(1 - 1.5n)
-125/3 = 1 - 1.5n
-128/3 = -1.5n
128/3 = 1.5n
log(128/3) = log1(.5n)
log(128/3) = n·log(1.5)
n = 9.3 ---> 10 terms
