Question 864619: Find the sum of the series: 12 - 6 + 3 - 3/2 + ...
Answer by richwmiller(17219)   (Show Source):
You can put this solution on YOUR website! S=t*(1 - r^n)/(1 - r)
where r=-1/3 t=12
S=12*(1 - 1/3^n)/(1 - (-1/3))
S = 9(1-3^(-n))
An infinite geometric series converges if its common ratio r satisfies –1 < r < 1.
Our ratio is -1/3 so it converges.
lim_(n->infinity) 9 (1-3^(-n)) = 9
S = 9 (1-3^(-n))
let's try a few numbers and see what happens
S=12*(1 - 1/3^10)/(1 - (-1/3)),
T=12*(1 - 1/3^20)/(1 - (-1/3)),
U=12*(1 - 1/3^30)/(1 - (-1/3))
V=12*(1 - 1/3^100)/(1 - (-1/3))
as expected
S=8.9998, T=9.0000, U=9.0000, V=9.0000
even as early as n=10 it is almost 9
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