2 Show that the following sequences are divergentSolutionAns

2. Show that the following sequences are divergent.

Solution

Answer 1)

Let me call it a_n, that is a_n = sin(n/2).

Consider the subsequence

a_k = a_(2n) = sin(2) = 0 for all k.

This subsequence converges to zero. Consider the subsequence

a_j = a_(4n+2) = sin((4n+2)/4) = sin((2n+1)/2) = 1 for all j.

This subsequence converges to 1.

As the sequence admits convergent subsequences with different limits, we can conclude that the sequence is not convergent.

that means the given sequance is divergent .

2 ) cosnider lim n --> infinity n pi / 2   =   pi/ 2 ( lim n --> infinity   n )   = pi / 2 * inifnity  

that means the given sequance is divergent.