Introduction to sequences

2010010306

Level: 
B
Which of following statements about the sequence \( \left( \frac{n-2}{n+1}\right)^{\infty}_{n=1} \) is true? \[\] (Help: A sequence is bounded below if all its terms are greater than or equal to a real number \(L\), which is called the lower bound of the sequence. Similarly, a sequence is bounded above if all its terms are less than or equal to a real number \(U\), which is called the upper bound of the sequence.)
one of lower bounds is \(-\frac12\), one of upper bounds is \(1\)
one of lower bounds is \(-\frac12\), upper bound does not exist
lower bound does not exist, one of upper bounds is \(1\)
there exists neither lower bound nor upper bound

2010010305

Level: 
B
Which of following statements about the sequence \( \left( \frac{n+3}{2n}\right)^{\infty}_{n=1} \) is true? \[\] (Help: A sequence is bounded below if all its terms are greater than or equal to a real number \(L\), which is called the lower bound of the sequence. Similarly, a sequence is bounded above if all its terms are less than or equal to a real number \(U\), which is called the upper bound of the sequence.)
one of lower bounds is \(\frac12\), one of upper bounds is \(2\)
one of lower bounds is \(\frac12\), upper bound does not exist
lower bound does not exist, one of upper bounds is \(2\)
there exists neither lower bound nor upper bound

2010010304

Level: 
B
We are given a sequence \( \left( \frac{n+5}{n+1}\right)^{\infty}_{n=1}\). What are the properties of this sequence?
decreasing and bounded above
increasing and not bounded below
neither increasing nor decreasing
increasing and bounded below
decreasing and not bounded above

2010010303

Level: 
B
We are given a sequence \( \left( \frac{2n+1}{n+3}\right)^{\infty}_{n=1}\). What are the properties of this sequence?
increasing and bounded
neither increasing nor decreasing
decreasing and bounded above
increasing and not bounded above
decreasing and not bounded above