This is NOT . There is no absolute value in .

Then both you and your book are wrong! For n= 10000000, say, , approximately, very close to 1 while if n= 1000001, , approximately, which is very close to -1.

My book says the sequence converges at -1.

This sequence doesnotconverge. It has asubsequence (n even) that converges to 1 and another (n odd) that converges to -1.