I have a question: what is lim (n--->infinity)= 1/(3+(-1)^n))? My opinion that this limit does not exist.
Are you asking about [tex]\lim_{n\rightarrow\infty}\frac{1}{3+(-1)^n}[/tex] perhaps? The equals sign in your post is confusing me. If so, are you familiar with the lim sup and lim inf? That would give you an easy direct proof: if lim sup = lim inf, that's the limit; otherwise, the limit does not exist.
manooch if n∈Z (Z=Integer) then we have two answer for equation 1) if n=Even then answer=1/4 2) if n=Odd then answer=1/2 if n∈R (R=Real) then equation is undefined for example: (-1)^1/2 does not exist.
I think you could use: Proposition 4 Every subsequence of a convergent sequence converges to the same limit. from: http://www.iwu.edu/~lstout/sequences/node3.html