Suppose all the terms of an increasing sequence a(n) lie between -4 and -10.  Is this enough info to determine anything about the limit of the seq.?Justify your answer

Expert Answers
kjcdb8er eNotes educator| Certified Educator

Yes. Because you know that the series is increasing, it cannot oscillate between two values. Therefore this series increases until it approaches the final value, or limit, or -4.

Note that if all you knew was that the values of the series were between -4 and -10, the answer would be different, since the series could alternate forever between two values, say -5 and -8.

neela | Student

We know that a monotonic nondecreasing  sequence which has an upper bound has a least upper bound.

So if  f(x)  is  nondecreasing and f(x) < k  for all x, then it has a least upper bound.

This implies that there is a least upper bound M such that

f(n) < n for al n >m.

This implies that for any e however small  M-e < f(x) < M for all n >m.

Therfore Lt n--> Lt(n) = M.

In the given case f(n) has the upper bound = -4.

f(n) is increasing (So the nondecreasing condition is satisfied).

Therefore f(n) has a least upper bound < -4.

So Lt {f(n) as x--> infinity} = M . Where -10 <M<-4.

The conditions are sufficient to say Lt f(n) = M as n --> infinity.