text: # Determine if a sequence {an} is monotonic, bounded, convergent
  
  1. Jan 12, 2008
  
  ### sara_87
  
  Determine whether the sequence {an} defined below is
  (a) monotonic
  (b) bounded
  (c) convergent and if so determine the limit.
  
  (1) {an}=(sqrt(n))/1000
  a) it is monotonic as the sequence increase as n increases.
  b) it's not bounded (but i'm not sure why)
  c) divergent since limit doesnt exit (tends to infinity)
  is this correct?
  (2) {an}=(-2n^2)/((4n^2)-1)
  i dont know how to answer parts (a) (b) and (c) to this one can someone help me please.
  thank you
  
  2. Jan 14, 2008
  
  ### Gib Z
  
  1) a) Correct.
  
  b) A sequence is bounded if |a_n| is less than some real number M, for all values of n. Does this help? Is there such a number M in this case?
  
  c) Correct. Also, an unbounded monotonic increasing sequence necessarily diverges to positive infinity. It may be a good exercise to prove why.
  
  2) a) Trick is to subtract a half and then add it again, like this;
  
  $$a_n = - \left( \frac{2n^2}{4n^2 -1} \right) = - \left( \frac{2n^2 - \frac{1}{2} }{4n^2 -1} + \frac{1}{2(4n^2-1)}\right) = - \left ( \frac{1}{2} + \frac{1}{2(4n^2-1)}\right)$$.
  
  Now use partial fractions, it should be much easier then =]
  
  PS - I noticed its been about one and a half days since you've posted this, and yet received no help. This is probably because you placed this in the wrong forum - Put it in the Homework Help forums (Calculus and Beyond) next time and you should get help much faster.
  
  3. Jan 14, 2008
  
  ### dodo
  
  A common trick with powers in a fraction is to divide up and down by the highest power, like this:
  $$a_n = -\frac{\frac{2 n^2}{n^2}}{\frac{4 n^2 - 1}{n^2}} = -\frac 2 {4 - \frac 1 {n^2}}$$​
  Now it's easy to see what the limit is.
  
  4. Jan 14, 2008
  
  ### Gib Z
  
  Shoot me that is a much easier way :(
explanation: The document contains a math question and a partial answer, followed by a detailed response from another user providing further explanation and guidance. The initial user asks about the properties of two sequences, and the responder provides answers and hints for the first sequence and a simpler method for the second sequence.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Determine if a sequence {an} is monotonic, bounded, convergent
  
  1. Jan 12, 2008
  
  ### sara_87
  
  Determine whether the sequence {an} defined below is
  (a) monotonic
  (b) bounded
  (c) convergent and if so determine the limit.
  
  (1) {an}=(sqrt(n))/1000
  a) it is monotonic as the sequence increase as n increases.
  b) it's not bounded (but i'm not sure why)
  c) divergent since limit doesnt exit (tends to infinity)
  is this correct?
  (2) {an}=(-2n^2)/((4n^2)-1)
  i dont know how to answer parts (a) (b) and (c) to this one can someone help me please.
  thank you
  
  2. Jan 14, 2008
  
  ### Gib Z
  
  1) a) Correct.
  
  b) A sequence is bounded if |a_n| is less than some real number M, for all values of n. Does this help? Is there such a number M in this case?
  
  c) Correct. Also, an unbounded monotonic increasing sequence necessarily diverges to positive infinity. It may be a good exercise to prove why.
  
  2) a) Trick is to subtract a half and then add it again, like this;
  
  $$a_n = - \left( \frac{2n^2}{4n^2 -1} \right) = - \left( \frac{2n^2 - \frac{1}{2} }{4n^2 -1} + \frac{1}{2(4n^2-1)}\right) = - \left ( \frac{1}{2} + \frac{1}{2(4n^2-1)}\right)$$.
  
  Now use partial fractions, it should be much easier then =]
  
  PS - I noticed its been about one and a half days since you've posted this, and yet received no help. This is probably because you placed this in the wrong forum - Put it in the Homework Help forums (Calculus and Beyond) next time and you should get help much faster.
  
  3. Jan 14, 2008
  
  ### dodo
  
  A common trick with powers in a fraction is to divide up and down by the highest power, like this:
  $$a_n = -\frac{\frac{2 n^2}{n^2}}{\frac{4 n^2 - 1}{n^2}} = -\frac 2 {4 - \frac 1 {n^2}}$$​
  Now it's easy to see what the limit is.
  
  4. Jan 14, 2008
  
  ### Gib Z
  
  Shoot me that is a much easier way :(
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and a partial answer, followed by a detailed response from another user providing further explanation and guidance. The initial user asks about the properties of two sequences, and the responder provides answers and hints for the first sequence and a simpler method for the second sequence.
  
  Label: question-and-answer