Question

whats the difference between below two limit notations

Answer 1

1) \[\lim \limits_{n\to \infty} (a_n)^{1/n}\]

2)
\[\lim ~\sup~ (a_n)^{1/n}\]

Answer 2

2) always exists for a sequence bounded above, but 1) may not.

Answer 3

thanks, my textbook says something similar but i really dont get the difference yet

Answer 4

Just try a few examples and see what you get.

Answer 5

i think supremum is the limit of a sequence so they both give same numbers right ?

\[\lim \limits_{n\to \infty} \frac{1}{n} = \lim \sup \frac{1}{n} = 0\]

Answer 6

another example

\[\lim \limits_{n\to \infty}1 = \lim \sup 1= 1\]

Answer 7

take an oscillating sequence

Answer 8

another example

\[\lim \limits_{n\to \infty} (-1)^n = DNE \\~\\\lim \sup (-1)^n= ?\] does this exist ?

Answer 9

yes

Answer 10

+1 ?

Answer 11

and inf = -1 ?
Oh ooh

Answer 12

in fact, limsup is the largest limit point of a (bounded) sequence

Answer 13

i suppose below should not exist

\[\lim\limits_{n\to \infty} \sup (-1)^n= ?\] does this exist ?

Answer 14

limsup exists for bounded sequences

Answer 15

limit may not

Answer 16

real quick question

\(\lim\sup \frac{1}{n}\) equals 1 or 0 ?

Answer 17

if i consider the sequence : {1, 1/2, 1/3, ... }

inf = 0
sup = 1

Answer 18

there is no difference in general , it only depend on the example or reason u have , the thing is sometimes when a sequence has a sup then its bounded above .. that make limit exist all the time

Answer 19

1 :)

Answer 20

whats diffference between sup and limsup ?

Answer 21

\[\limsup_{n\to\infty}x_n := \lim_{n\to\infty}\Big(\sup_{m\geq n}x_m\Big)\]

\[\limsup_{n\to\infty}\frac{1}{n} = \lim_{n\to\infty}\Big(\sup_{m\geq n}\frac{1}{m}\Big)\]

\[=\lim_{n\to\infty}\frac{1}{n}=0\]

Answer 22

\(\lim\limits_{n\to \infty} \sup~x_n\) considers only the tail of sequence @Zarkon ?

Answer 23

\[\limsup_{n\to\infty}(-1)^n= \lim_{n\to\infty}\Big(\sup_{m\geq n}(-1)^m\Big)\]

\[=\lim_{n\to\infty}1=1\]

Answer 24

sup means the least greatest value of the sequence
inf means the greatest smallest value , consider this open interval:-

Answer 25

sup is the least upper bound.
inf is the greatest lower bound.

Answer 26

Ohk, so the sequence need not converge to sup or inf right ?

Answer 27

it could converge to some number in between