Determine the sum of the series 1, 1/2, 1/4, 1/8 .......... t(14)?

1 Answer
Jun 7, 2016

sum_{i=0}^oo 1/2^i = 2 or
sum_{i=0}^14 (1/2)^i ==32767/16384 = 1.9999389648437500000

Explanation:

sum_{i=0}^oo 1/2^i = sum_{i=0}^oo (1/2)^i

We know that

(1-x^{n+1})/(1-x) = 1 + x + x^2+ cdots + x^n and that

for abs(x)<1 we havelim_{n->oo}(1-x^{n+1})/(1-x)=1/(1-x)

Now 1/2 < 1 so

sum_{i=0}^oo (1/2)^i = 1/(1-1/2) = 2

also if the sumation is done from i=0 to i = 14
then

sum_{i=0}^14 (1/2)^i = (1-(1/2)^{15+1})/(1-(1/2)) =32767/16384 = 1.9999389648437500000