Least upper bound, greatest lower bound, limit superior, limit inferior 2.20. Find the (a) 1.u.b., (b) g.l.b., (c) lim sup ( lim ), and (d) lim inf (lim) for the sequence 2, -2, 1, -1, 1,-1, 1, -1,.... (a) l.u.b. = 2, since all terms are less than equal to 2, while at least one term (the 1st) is greater than 2- e for any e > 0. (b) g.l.b. = -2, since all terms are greater than or equal to -2, while at least one term (the 2nd) is less than -2 + e for any e > 0. (c) lim sup or lim = 1, since infinitely many terms of the sequence are greater than 1- e for any e > 0 (namely, all 1's in the sequence), while only a finite number of terms are greater than 1 + e for any e >0 (namely, the 1st term). (d) lim inf or lim =-1, since infinitely many terms of the sequence are less than -1 + e for any e > 0 (namely, all -1's in the sequence), while only a finite number of terms are less than -1 - e for any e > 0 (namely, the 2nd term).

Least upper bound, greatest lower bound, limit superior, limit inferior 2.20. Find the (a) 1.u.b., (b) g.l.b., (c) lim sup ( lim ), and (d) lim inf (lim) for the sequence 2, -2, 1, -1, 1,-1, 1, -1,.... (a) l.u.b. = 2, since all terms are less than equal to 2, while at least one term (the 1st) is greater than 2- e for any e > 0. (b) g.l.b. = -2, since all terms are greater than or equal to -2, while at least one term (the 2nd) is less than -2 + e for any e > 0. (c) lim sup or lim = 1, since infinitely many terms of the sequence are greater than 1- e for any e > 0 (namely, all 1's in the sequence), while only a finite number of terms are greater than 1 + e for any e >0 (namely, the 1st term). (d) lim inf or lim =-1, since infinitely many terms of the sequence are less than -1 + e for any e > 0 (namely, all -1's in the sequence), while only a finite number of terms are less than -1 - e for any e > 0 (namely, the 2nd term).

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Least upper bound, greatest lower bound, limit superior, limit inferior
2.20.
Find the (a) 1.u.b., (b) g.l.b., (c) lim sup ( lim ), and (d) lim inf (lim) for the sequence 2, -2, 1, -1, 1, -1, 1,
-1,....
(a) l.u.b. = 2, since all terms are less than equal to 2, while at least one term (the 1st) is greater than 2 - e for
any e > 0.
(b) g.l.b. = -2, since all terms are greater than or equal to -2, while at least one term (the 2nd) is less than
-2 + e for any e > 0.
(c) lim sup or lim = 1, since infinitely many terms of the sequence are greater than 1 – e for any e > 0
(namely, all 1's in the sequence), while only a finite number of terms are greater than 1 + e for any e > 0
(namely, the 1st term).
(d) lim inf or lim = -1, since infinitely many terms of the sequence are less than -1+ e for any e > 0 (namely,
all -l's in the sequence), while only a finite number of terms are less than -1 – e for any e > 0 (namely,
the 2nd term).

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