MATH 227 - True or False

If a sequence isn’t monotonic, it’s also not bounded

False because it’s possible for a sequence to not be monotonic, but be bounded

If the limit of a sequence goes to a finite number, then the sequence diverges

False because if a sequence goes to a finite number, then the sequence converges

A series is convergent if the sequence of partial sums is a convergent sequence (don’t explain)

True

Which three tests require the series to be positive

1. Integral Test

2. Direct Comparison Test

3. Limit Comparison Test

Pick the right choices and explain why: If an alternating series converges/diverges, then the remainder R_n ≥ / ≤ b_{n + 1}

If an alternating series converges, then the remainder R_n ≤ b_{n + 1} because of the alternating series estimation theorem