Determine with limitrules if following sequence convergence, if yes then calculate its limit

$$b_n := \left\{
\begin{array}{l l}
\frac{1+n}{n} & \quad \text{falls $n$ is odd }\
\frac{1-n}{n} & \quad \text{if $n$ is even}\
\end{array} \right.$$

for $$n\geq 1$$

Question

Determine with limitrules if following sequence convergence, if yes then calculate its limit

$$b_n := \left\{ 
  \begin{array}{l l}
    \frac{1+n}{n} & \quad \text{falls $n$ is odd }\
    \frac{1-n}{n} & \quad \text{if $n$ is even}\
  \end{array} \right.$$

for $$n\geq 1$$

experimentx · Answer

the sequence is an oscillating sequence ... it does not converge.

anonymous · Answer

i think the first one is not convergennce because  1+n/n  > 1 
second onve convergence because the solution is less then -1

anonymous · Answer

and how to show it in exam in mathematical way ?

experimentx · Answer

no ... when n-> infinity, the above sequence converges to 1 while the later sequence converges to -1
you cannot have two different limit point for same sequence. so it does not converge.

anonymous · Answer

ok thank you experimentX

experimentx · Answer

yw .. you know (-1)^n does not converge ... this is quite similar to it.

anonymous · Answer

you mean for the second sequence?

anonymous · Answer

i know $$-1^{n}$$ alone it does not convergence