Understand the problem
$ a_n=\begin{cases}
2+\frac{\{-1\}^{\frac{n-1}{2}}}{n}, & \text{if n is odd}\\
1+ \frac{1}{2^n}, & \text{if n is even}
\end{cases}$
Which of the following is true?
(a) sup {\(a_n|n \in \mathbb{N}\)}=3 and inf {\(a_n|n \in \mathbb{N}\)}=1
(b) lim inf (\(a_n\))=lim sup (\(a_n\))=\(\frac{3}{2}\)
(c) sup {\(a_n|n \in \mathbb{N}\)}=2 and inf {\(a_n|n \in \mathbb{N}\)}=1
(d) lim inf (\(a_n\))= 1 lim sup (\(a_n\))=3
Start with hints
Hint 1:$a_n=\begin{cases}
2+\frac{\{-1\}^{\frac{n-1}{2}}}{n}, & \text{if n is odd}\\
1+ \frac{1}{2^n}, & \text{if n is even}
\end{cases}$
Now the limit points of this set are those points which the set does not attain.So, they might be the sup and inf which are not attained by this set.
Basically sup(\(a_n\))= max{ limit points, \(a_n\) | n \(\in\) \(\mathbb{N}\)}
Limit points are \(2,1\) and \(a_1= 2+1=3, a_3= 2- \frac{1}{3} ; a_5= 2+\frac{1}{5} \)
\(a_0= 1+1=2 , a_2= 1+ \frac{1}{4} , a_3= 1+\frac{1}{8} \)
Now you can calculate the supremum?
Hint 2:From the observation of Hint 2 we have
sup \(a_n\)= max \(\{2,1,3,2\}=3 \)
Similarly, inf \(a_n\)= min\(\{\) limit points, \(a_n | n \in \mathbb{N}\}\)
Can you calculate that by yourself?Hint 3:inf \(a_n\)= min {2,1,2 -\(\frac{1}{3}\)}=1
So, option A is correct.
Now there is another question regarding lim sup and lim inf. We can observe that we have mainly \(3\) subsequences , corresponding to
\( n\) is even; \(n=2k\)
\(n\)= \(4k+1\)
\(n=4k+3\)Can you calculate the corresponding subsequences and their limits?
Hint 4:For \(n=2k\) we have
\(a_{2k}=1+ \frac{1}{2^{ek}} \longrightarrow 1 \) ask
For \(a_{4k+1}= 2+ \frac{1}{4k+1} \longrightarrow 2\) ask
\(a_{4k+3}= 2-\frac{1}{4k+3} \longrightarrow 2\) ask
So, lim sup \(a_n\)=max\(\{1,2\}=2\)
Lim inf \(a_n\)=min\(\{1,2\}=1\)
Therefore, Option C is also correctConnected Program at Cheenta
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