Help find Lim Sup and Lim Inf of the following sequences:

Question

I'll be honest just looking for confirmation of my answers. I have been given the sequences $S_n= \pi+(-1)^n$, $A_n=\pi+(-0.5)^n$, $B_n=\pi+(-2)^n$.

I am asked to find Limsup and LimInf for each sequence and I have tried by simply observing that there are only 2 subsequences for each sequence those being $S_{2n}$ and $S_{2n-1}$ which when finding the limits I find the results:

$$LimSup(S_n)=\pi+1, LimInf(S_n)=\pi-1$$

$$LimSup(A_n)=LimInf(A_n)=\pi$$

$$LimSup(B_n)=+∞, LimInf(B_n)=-\infty$$

Am I doing this the correct way for sequences that oscillate between positive and negative?

Answer 1

The answers are right. Remember the definition of limsup and liminf. You need to find the set of limits of all converging subsequences first( including those “converge” to infinity). Then find the least upper bound(sup) and the greatest lower bound(inf) of the set. These two bounds are limsup and liminf, respectively.