Thread: limsup and liminf

I have to find the limit superior and the limit inferior of these sequences:

a)
n + [(-1)^n](2n+1)
---------------- It appears limsup = 3 and liminf = -1.
n

b)
(an)^(1/n) where an = 1/(2^[(n+1)/2]) if n is odd and 1/(3^(n/2)) if n is even. What are the limsup and liminf of these? They seem less obvious.

Thanks in advance for any help.

Originally Posted by MKLyon

I have to find the limit superior and the limit inferior of these sequences:

a)
n + [(-1)^n](2n+1)
---------------- It appears limsup = 3 and liminf = -1.
n

i agree

b)
(an)^(1/n) where an = 1/(2^[(n+1)/2]) if n is odd and 1/(3^(n/2)) if n is even. What are the limsup and liminf of these? They seem less obvious.

Thanks in advance for any help.

we have:

thus

can you take it from here?

consider when as well as when

So 1/sqrt(2) and 1/sqrt(3)?

Thanks for responding.

Originally Posted by MKLyon

So 1/sqrt(2) and 1/sqrt(3)?

Thanks for responding.

did you consider when n = 1? what happens then?