1. 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.

2. 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: $a_n = \left \{ \begin{array}{lr} \frac 1{2^{(n + 1)/2}} , & \mbox{ for } n \mbox { odd} \\ & \\ \frac 1{3^{n/2}} , & \mbox{ for } n \mbox{ even} \end{array}\right.$

thus $(a_n)^{\frac 1n} = \left \{ \begin{array}{lr} \frac 1{2^{(n + 1)/2n}} , & \mbox{ for } n \mbox { odd} \\ & \\ \frac 1{\sqrt{3}} , & \mbox{ for } n \mbox{ even} \end{array}\right.$

can you take it from here?

consider when $n = 1$ as well as when $n \to \infty$

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

Thanks for responding.

4. Originally Posted by MKLyon
So 1/sqrt(2) and 1/sqrt(3)?

Thanks for responding.
did you consider when n = 1? what happens then?