Limit Inferior and Limit Superior

[veronicak5678]

Prove that for any positive sequence a_n of real numbers
lim inf (a _(n+1) / a_n) <= lim inf (a_n)^(1/n) <= lim sup (a_n)^(1/n)
<= lim sup(a_(n+1) / a_n).
Give examples where equality does not hold.

Is lim sup always >= lim inf? I am having trouble understanding these concept and proving things about them without specific numbers.

 

[mathman]

For your second question, the answer is obviously yes. Inf is always ≤ sup, so this relationship will hold as you take limits.

An example for the first question, the following sequence will work:
a_n = 2ⁿ for even n.
a_n = 1/2 for odd n.

lim inf a_n+1/a_n = 0
lim inf a_n^1/n = 1
lim sup a_n^1/n = 2
lim sup a_n+1/a_n = ∞

 

[veronicak5678]

Could you please show how you got those values? I am obviously confused about the definition. My book isn't making much sense to me.

 

[mathman]

I suggest you get a better handle on the meanings of inf, sup, lim inf, and lim sup.

For the ratio, when n is odd the value is 2ⁿ⁺², and when n is even the value is 2^-n-1, so as n becomes infinite, the lim inf = 0 and the lim sup is infinite.

 

[alexfloo]

http://en.wikipedia.org/wiki/Limit_superior_and_limit_inferior

The graphic on this page is a useful illustration. (Wikipedia has an unfortunate habit of getting extremely technical, and not separating the complex concepts form the simpler ones, so if the rest of the page isn't all that helpful, don't be discouraged.)

Basically, (and these definitions are not rigorous) the lim sup is the is the "smallest value that the sequence is *eventually* not bigger than" and the lim inf is the "largest value that the sequence is eventually not smaller than." Compare these descriptions with the picture, and hopefully it'll make more sense.