Determine if the sequence an*= {3/4, 1/8, -1/2, -9/8...} converges or diverges

Question

then determine if the associated series 3/4 +1/8+ (-1/2)+(-9/8)... converges or diverges
*the n is in subscript

Answer 1

hard to say. There is no common ratio, and I see no useful pattern in the fractions.

The denominators are all powers of 2, but they are 2,3,1,3 ...
If we use a common denominator of 8, the numerators are 6,1,4,9 ...

I guess you could separate it into two geometric sequences, with
a = 3/4 and r = 1/6
and
a = -1/2 and r = 9/4
Those would both converge, and the sums are easy to calculate.

Other than that, I got nothing.

Answer 2

Going with ooblecks idea of finding a common denominator of 8, we get

{3/4, 1/8, -1/2, -9/8...}
= {6/8, 1/8, -4/8, -9/8...} , which is an arithmetic sequence, with common difference of -5/8

What do you know about convergence and divergence of arithmetic sequences ?

Answer 3

1. D) The sequence diverges; the series diverges
2. D) geometric, divergent
3. C & E) a geometric sequence with r = (3/5) ; a geometric sequence with r = -(1/6)

might be different for non-honors

Answer 4

honors student is correct