Find the sum of the geometric series or state that the sum does not exist.?

Question

(1) 8 - 4 + 2 - ...

(2) 1+ 1.5 + 2.25 + ...

Answer 1

(1) No idea.

(2) Infinity. Imagine a 1.5^0 + 1.5^1 + 1.5^2 + ... + 1.5^9999 + ... + 1.5^[infinity].

Answer 2

1)

8 - 4 + 2 -------------

8 + 8*(-1/2) + -4(-1/2)

here a = 8: r = -1/2

Sum upto infinity = a/(1-r) when r < 1

since r is less than 1

S(∞) = 8/[(1 -(-1/2)]

S(∞) = 8/3/2 = 16/3

2)

1 + 1.5 + 2.25

1 + 1*1.5 + (1.5)*(1.5)----------------

here a = 1 and r = 1.5

since r >1, the series is not convergent and sum does not exist

Answer 3

r = 0.5 and the formula for an inf geom series is

t1/(1-r)

sum = 8 /(1.5) = 5.333