Limit superior and limit inferior

#limit superior#limit inferior#sequences

Statement

Compute $\limsup$ and $\liminf$ of $u_n=(-1)^n\left(1+\frac1n\right)$ .

Hint

Consider even and odd subsequences.

Show solutionHide solution

Even. $u_{2k}=1+\frac{1}{2k}\to1$ .

Odd. $u_{2k+1}=-(1+\frac{1}{2k+1})\to-1$ .

Conclusion. $\limsup u_n=1$ , $\liminf u_n=-1$ ; the sequence has no limit.