Question

could you use a(n) = (-1)^n to prove this?

4 ( 3pts) What could go wrong in the ratio test from Theorem 7.6 it an is allowed (encouraged] to be negative? (Hint: try an alternating sequence.)

Answer 1

if we take an sequence a(n) = -1 , 1 ,-1, 1 ,-1.........

this seuence is not monotonicity since if we use suppose sequence 1,1,1,1......

then according to theorem 2    will be true and it monotonocially iincreasing function

ak+1 >=

but if a(n) = -1, 1, -1, 1, -1........

since a2/a1 = -1, a3/a2 = -1, a4/a3= -1....... which is not >=1

then theorem 2 will false for every k so it fails monotonically test

if we allow any negative number in sequence then it fails ratio test look another sequence

1,2,3,4,5,-6,7,8,9,10........

then a2/a1=2>=1

a3/a2=3/2=1.5>=1

and so on but a6/a5=-6/5=-1.2 which fails>=1 similarly a7/a6=7/-6=-1.16 which fails >=1

so it fails monotonically test

since if we divide a positive number with a negative number or vice versa then result will always negative so condition don't hold true of ratio test for all k condition is not satisfy so given sequence will not monotonic.....