Which of the following sequences are bounded n2 13n 1 lnn

Which of the following sequences are bounded? {n^2 + 1/3n + 1} {ln(n + 1)} {1 - (-1)6n/n} {1/sin(n-1)} {e^-n}

Solution

A) the given sequence is bounded below but not above as n^2 +1 > 3n+1 for n 4 hence as we increase the value of n , value of sequence increases thus not bounded above .

B) log (n+1) is again not bounded as with the increase in value of n it twnds to infinity . Hence not bounded above.

C) (1+(-1)^n/n) is a bounded sequence as all its value lie between 0 and 2 . Thus bounded below by 0 and above by 2.

D) 1/sin(n-1) is bounded as 0 |sin(n-1)| n-1 thus by taking reciprocal we get that it is bounded .

E) e^-n is bounded as if n tends to infinity e^-n tends to 0 and is bounded by 1 also as e^0 is 1 . Thus e-n is boundec