If we negate every other term of the sequence (-1)^-n what would happen to the sequence?

Question

It seems obvious that without negating every other term, the sequence alternates btwn -1 and 1. So if you negate every other term, it would just be -1 repeatedly. Right? Would you say the sequence converges to -1?

Answer 1

Yes, (-1)^(-n) would just be the alternating sequence of 1 and -1 (where it's 1 if n is even, and -1 if n is odd). So "negating every other term" would just be the sequence "1, 1, 1, .." or "-1, -1, -1,..." It obviously depends on what value of n you're using to start your sequence (could be n=0 or n=1), and where you want to start counting "every other term". Regardless, if it's the same number repeated, then yes the new sequence converges to that number.

Answer 2

Negating every other term gives you a string of 1's or -1's, depending on where you start. The series, or sum of the sequence becomes n or -n, and does not converge in either case. Strictly speaking convergence or divergence of the sequence becomes meaningless since there is no difference in terms to get larger or smaller.

Answer 3

depends on if you're negative the even terms or odd terms.

could converge to 1 or -1

Answer 4

Your series does not converge to any distinct value because
|r| =1. Therefore it is divergent.